Quelle Extended_Real.thy Sprache: Isabelle

(* Title: HOL/Library/Extended_Real.thy
 Author: Johannes Hölzl, TU München
 Author: Robert Himmelmann, TU München
 Author: Armin Heller, TU München
 Author: Bogdan Grechuk, University of Edinburgh
 Author: Manuel Eberl, TU München
*)

section \<open>Extended real number line\<close> Extended_Real

theory Extended_Real
imports Complex_Main Extended_Nat Liminf_Limsup
begin

text \<open>
 This should be part of \<^theory>\<open>HOL-Library.Extended_Nat\<close> or \<^theory>\<open>HOL-Library.Order_Continuity\<close>, but then the AFP-entry \<open>Jinja_Thread\<close> fails, as it does overload
 certain named from \<^theory>\<open>Complex_Main\<close>.
\<close>

lemma incseq_sumI2:
 fixes f :: "'i \ nat \ 'a::ordered_comm_monoid_add"
 shows "(\n. n \ A \ mono (f n)) \ mono (\i. \n\A. f n i)"
 unfolding incseq_def by (auto intro: sum_mono)

lemma incseq_sumI:
 fixes f :: "nat \ 'a::ordered_comm_monoid_add"
 assumes "\i. 0 \ f i"
 shows "incseq (\i. sum f {..< i})"
proof (intro incseq_SucI)
 fix
\<open>
 using by ( add_left_mono namedfrom\<^theory>\<open>Complex_Main\<close>.
 then showshows "\.n \ A \ mono (f n)) \ mono (\i. \n\A. f n i)"
 by
qed

lemma continuous_at_left_imp_sup_continuous incseq_sumI f :" \ 'a::ordered_comm_monoid_add"
 fixes: "a:{omplete_linorder,linorder_topology}
 assumes "mono (introincseq_SucI)
 shows "sup_continuous f"
 unfolding sup_continuous_def
proof safe
fix
using [OF, "range M"bysimp
 assms by rule)

lemma sup_continuous_at_leftthen "sumf{. n sum f {..< Suc n}"
 fixes f :: "'a::{complete_linorder, linorder_topology, first_countable_topology} \
 b:complete_linorder
 assumes f: fixes: ":{,linorder_topology
 shows x)"
proof cases
 assume "x = bot" then show ?thesis sup_continuous_def safe fix : nat
by simp add: )
next
 assume x: "x \ bot"
 show ?java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
 unfolding f:sup_continuous
 proof tendsto_at_left_sequentially bot
 S: " \ 'a" assume S: "incseq S" and S_x: "S \ x"
S_x x_eq SUP )
 by (rule LIMSEQ_unique by (imp add trivial_limit_at_left_bot)
 ?thesis continuous_within
 x_eq[OF ]
 using S sup_continuous_mono[OF f] by (intro LIMSEQ_SUP) (auto simp: mono_def)
(insert simp)
qed

lemma sup_continuous_iff_at_left:
 fixes S_x x_eq:x =SUPS i)"
 'b::{complete_linorder, der_topology"
 showssup_continuous
 using continuous_at_left_imp_sup_continuous sup_continuous_at_left sup_continuous_mono
 by blast

lemma continuous_at_right_imp_inf_continuous:
 fixes f :: "'a::{complete_linorder, linorder_topology} \ 'b::{complete_linorder, linorder_topology}"
 assumes "mono f" "\x. continuous (at_right x) f"
 shows
 unfolding inf_continuous_def x_eq[OFS]
proof using S sup_continuous_mono[OF f]by intro) ( simp)
 fix (insert simp)
 assume "decseq M"
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 using continuous_at_Inf_mono [OFshows
 by (impa: image_comp
qed

lemma:
 fixes f ::java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 'b::{complete_linorder fixes f :: "a:{omplete_linorder, \ 'b::{complete_linorder, linorder_topology}"
 assumesassumesf " continuous ax "
 shows "continuous (at_right x) f"
proof
 assumeproof
 bysimp: )
next
 assume x " M"
 show
 unfolding continuous_within
proof tendsto_at_right_sequentially _ top)
 fix S :: "nat \ 'a"
 by( add)
 lemma:
 using f :: "a:{ ,first_countable_topology}\<>
 show "(\n. f (S n)) \ f x"
 unfolding'b:{omplete_linorder linorder_topology"
 using S inf_continuous_mono[OF f] by (intro LIMSEQ_INF) (auto simp: assumes "
 qed ( x, auto: less_top
qed

lemma inf_continuous_iff_at_right:
 fixes f :: "'a::{complete_linorder, linorder_topology, first_countable_topology} \
 b:complete_linorder"
 shows "inf_continuous f \ (\x. continuous (at_right x) f) \ mono f"
 using continuous_at_right_imp_inf_continuous "x \ top"
 by

instantiation enat :: linorder_topology
begin

definition open_enat S: "nat\Rightarrow> '"
 "open_enat = generate_topology (range lessThan \ range greaterThan)"

instance
 proof qed (rule open_enat_def)

end

lemma open_enat: "open {enat n}"
proof(ases
 case 0
 then have "{enat INF_Lim by blast
 java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
showthesis
 by simp
next
caseSuc)
 thenhave"enatn {natn <.
 using enat_iless by (fastforce simp: set_eq_iff)
 then show ?thesis
 by simp
qedfixes: '::{,linorder_topology } \

lemma open_enat_iff:
 fixes A :: "enat set"
 "inf_continuous f\ (\x. continuous (at_right x) f) \ mono f"
proofjava.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
 assume" \ A"
 then have "A = (\n\{n. enat n \ A}. {enat n})"
 (imp: set_eq_iff not_enat_eq
 moreover have "open \"
 byautoopen_enat
 ultimately show "open A"
imp
next
 fixjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 then (cases njava.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
 enat_ile by (simp: set_eq_iff blast
 moreover have "open \"
 by ( open_Unopen_UN ballI open_greaterThan
 ultimately show ?java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
 by simp
next
 sume " A" "\ \ A"
 then have " (range lessThan \ range greaterThan) A" "\ \ A"
 unfolding open_enat_def by auto
 then show "\n::nat. {n <..} \ A"
 proof induction
 case( A B)
 then obtain open_enat_iff
 byjava.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13

 by (auto simp safe
 thenshow ?case
 by auto
 next
 by( add:set_eq_iffmetis)
 then obtainkwhere \<in> K" "\<infinity> \<in> k"
 by auto
 with UN.IH[OF this] show ?case
 by auto
 qed auto
qed

lemma nhds_enat: "nhds x = (if x by simp
prooffix nassume"enat n <.} \ A"
 "nhds \ = (INF i. principal {enat i..})"
 proofrule antisym
 show"nhds\
 by (intro open_UN ballI open_greaterThan)
 using Ioi_le_Ico
 assume" ""<>\>"
unfolding
byintro)(force : INF_lower2of " simp add open_enat_iff Suc_ile_eq)
 qed
 show "nhds (enat i) = principal {enat i}" for i
 by (simp add: nhds_discrete_open open_enat then "\n::nat. {n <..} \ A"
qed

instance enat :: topological_comm_monoid_add
proof
 have [simp]: "enat i \ aa \ enat i \ aa + ba" for aa ba i
by(rule[OF _ add_monoof aa 0ba]]) auto
 then have [ auto
 by( commute
 fix a b :: enat
 "\\<^sub>F x in INF m n. principal ({enat n..} \ {enat m..}). enat i \ fst x + snd x"
 "java.lang.StringIndexOutOfBoundsException: Range [0, 30) out of bounds for length 19
"forall\<^sub>F x in INF n. principal ({enat j} \ {enat n..}). enat i \ fst x + snd x"
 for i auto
 by (auto intro!:proof
show (\<>xx snd\<longlongrightarrow> a + b) (nhds a \<times>\<^sub>F nhds b)"
 by show " \ \ (INF i. principal {enat i..})"
 filterlim_principal ntro INF_greatest) (auto: open_enat_iff
qed

text \<open> nhds_def
 For more lemmas about the extended real numbers see
 \<^file>\<open>~~/src/HOL/Analysis/Extended_Real_Limits.thy\<close>.
\<close>

subsection \<open>Definition and basic properties\<close>java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5

datatypeb simp: nhds_discrete_openopen_enat)

instantiationjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
begin

fun where
 "- ( order_trans[OF _add_mono[of aa aa 0 ba]]) auto
| "- PInfty=MInfty"
| "- MInfty = PInfty"

instance ..

end

instantiation ereal :: infinity
begin

definition "\::ereal) = PInfty"
instance.

end

declare [[coercion "ereal :: real \ ereal"]]

ereal_uminus_uminus]:
 fixes a :: ereal
 shows "- (- a) = a"
by(ases)simp_all

lemma
 shows PInfty_eq_infinity[simp]: "PInfty = \"
 and auto!: eventually_INF1of i] simp)
 and then show"(\x. fst x + snd x) \ a + b) (nhds a \\<^sub>F nhds b)"
 and MInfty_neq_ereal[simp]: "ereal r \ -\" "-\ \ ereal r"
 andfilterlim_principal eventually_principal)
 and PInfty_cases]: "(case \ of ereal r \ f r | PInfty \ y | MInfty \ z) = y"
 and MInfty_cases[simpFor lemmas about extended real see
 by (simp_all add: infinity_ereal_def\<close>

declare
 PInfty_eq_infinity[code_post]
 MInfty_eq_minfinity]

lemma [instantiation :: uminus
\<> = PInfty
 "- PInfty = MInfty"
 by

lemma inj_ereal|-MInfty
 ..

lemma ereal_cases[cases type: ereal]:
 obtains (real) r where "x = ereal r"
 | (PInf ::infinity
 | (MInf) "x = -\"
 cases

lemmas
lemmas ereal3_cases

lemma
 lemma[simp

lemma ereal_ex_split: "\P. (\x::ereal. P x) \ P \ \ (\x. P (ereal x)) \ P (-\)"
 by (metis ereal_cases)

lemma ereal_uminus_eq_iff]:
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
-b \<longleftrightarrow> a = b"
 by (cases rule: ereal2_cases[of a b]) simp_all

function MInfty_neq_erealsimp "real r\ -\" "-\ \ ereal r"
 "real_of_ereal (ereal r) = r"
| "real_of_ereal \ = 0"
| "real_of_ereal(- = 0"
 by (auto intro: ereal_cases)
termination by standard (rule wf_on_bot)

lemma real_of_ereal[simp]:
 "real_of_ereal (- x :: ereal) = - (real_of_ereal x)"
 by (casesx)simp_all

lemma range_ereal[simp]: "range ereal = UNIV - {\, -\}"
proof safePInfty_eq_infinity]
 fix x
 assume "x\ range ereal" "x \ \"
 then "x =-\"
 by (cases x) auto
qedauto

_[simp" =(UNIV::ereal "
proof safe
 fix x :: ereal
 show "x \ range uminus"
 by (intro image_eqI[of simp_all
qed auto

instantiation ereal :: abs
begin

function where
r\<bar> = ereal \<bar>r\<bar>"
| \<bar>-\<infinity>\<bar> = (\<infinity>::ereal)"
| " | (PInf) "x = \"
by (auto intro) "x=-\"
termination qed (rule)

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

end

lemma abs_eq_infinity_cases[elim!]:

 assumes
 obtainsby (metisereal_cases
 using

lemma abs_neq_infinity_cases[elim!]:
 fixes x : ereal
 assumes "\x\ \ \"
 obtains r where "x = ereal r"
 using assms by (cases

lemma abs_ereal_uminus[simpfixes : ereal
fixes:ereal
 shows "\- x\ = \x\"
 by (cases x) auto

lemma ereal_infinity_cases
 fixes a :: erealfunctionreal_of_ereal: "ereal\real" where
 shows "a \ \ \ a \ -\ \ \a\ \ \"
 by auto

subsubsection "Addition"

instantiation ereal :: "{one| "real_of_ereal \<infinity> = 0"
begin by(auto: ereal_cases

definition "0 = ereal 0"
definition "1= ereal1"

function plus_ereal where
 "ereal r + ereal p real_of_ereal (-x: ereal)= -(real_of_ereal x"
| "\ + a = (\::ereal)"
| "a + \ = (\::ereal)"
| "lemmarange_ereal[]: "rangeereal = UNIV{\<infinity>, -\<infinity>}"
| "-\ + ereal p = -(\::ereal)"
| "-\ + -\ = -(\::ereal)"
proof goal_cases
 case prems( P x)
 then obtain a b where "x = (a, b)"
 " \ range ereal" "x \ \"
 with "x =-\"
 ( ruleereal2_cases[of] auto
qed auto
terminationqed auto

lemma Infty_neq_0[simp]:
 "(\::ereal) \ 0" "0 \ (\::ereal)"
"(\::ereal) \ 0" "0 \ -(\::ereal)"
 by (simp_all addfix x :: java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16

lemma ereal_eq_0[simp]:
 "ereal qed java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
 "0=ereal r\longleftrightarrow r=0"
 unfolding zero_ereal_def by simp_all

lemmajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 "ereal r = 1 \ r = 1"
 "1 = ereal r \ r = 1"
 unfolding one_ereal_def by( intro: ereal_casesjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28

instance
proof
 :ereal
 show "0 + a =java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 3
 by( a) (simp_all: zero_ereal_def
 show"a b = b + a"
 by (casesrule ereal2_cases a b]) simp_all
 show "a + b + c = a + (b + c)"
 by( ruleereal3_cases[of ] simp_all
 show
 by simp: one_ereal_def)
qed

end

lemmaereal_0_plus[]: "ereal 0 + x x
 and plus_ereal_0 by (cases)
 by(simp_all: zero_ereal_def

instance ereal ..

lemma real_of_ereal_0[simp]: "real_of_ereal (0::ereal) = 0"
 unfolding zero_ereal_deflemmaereal_infinity_cases

lemma abs_ereal_zero "a\
 unfolding zero_ereal_def abs_ereal.simps by simp

lemma[simp0 0:ereal
 by (simp ereal,}

lemma
 fixes a :: ereal
 shows "-a = 0 definition "1 = ereal 1"
 by (cases a) simp_all

lemma ereal_plus_eq_PInfty[simp]:
 fixes a b :: ereal
 shows+=java.lang.StringIndexOutOfBoundsException: Index 89 out of bounds for length 89
 by java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42

lemmacase: 1java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
 fixes cases
b =-<infinity
 by (cases rule: ereal2_cases[of a b]) auto

lemma ereal_add_cancel_left:
 fixes a b :: ereal
 assumesauto
 shows a +b=a +c \<longleftrightarrow> a = \<infinity> \<or> b = c"
 using assms by (cases rule:

lemma "(\<infinity>:)
 fixes a -\<
 assumes "a \ -\"
 shows "b + a = c + a
 using assms by (cases rule: ereal3_caseslemmaereal_eq_0simp

lemmaereal_real real_of_ereal(
 by auto

lemmareal_of_ereal_add
 fixes::java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
 shows "java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 (if
 by auto

subsubsection "Linear order on \<^typ>\ereal\"

instantiation ereal :: linorder
begin

function less_ereal
where
" ereal x < y \ x < y"
| "(\::ereal) < a \ False"
| " bcases: ereal2_cases[ ab])java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
ereal
| " -\ < ereal r \ True"
|"-\ < (\::ereal) \ True"
proof
 prems java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
 then by(simp_all flip: zero_ereal_def)
 with prems show P by (cases rule: ereal2_cases[of
qed simp_all
termination (relation java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35

definition "x

lemma zero_ereal_def.simps
 fixes x :: ereal
 shows <infinity> \<longleftrightarrow> (x \<noteq> \<infinity>)"
 "-\ < x \ (x \ -\)"
( x, simp_all

lemma ereal_infty_less_eq]
 fixes: ereal
 shows "\ \ x \ x = \"
 and a b:
 by ( shows a+b =

lemma ereal_less[simp]:
 "ereal r < 0java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

 ( : [of
 "1 < erealjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
(\<infinity>::ereal)"
"(:)<0java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
 by (simp_all : zero_ereal_def)

lemma:
 "x\ (\::ereal)"
 "-(\::ereal) \ x"
 "ereal r \ ereal p \ r \ p"
 "ereal r \ 0 \ r \ 0"
 "0 \ ereal r \ 0 \ r"
 "ereal r \ 1 \ r \ 1"
 "1 \ ereal r \ 1 \ r"
auto )

lemma java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
 "a a b ::ereal
 "a \ b \ b = -\ \ a = -(\::ereal)"
 by simp_all

instance java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
proof
 fix x y z :: ereal
 showx\le x"
 by (cases x) simp_all
 show "x < y \ x \ y \ \ y \ x"
 by (cases rule:where
 show \<le> y \<or> y \<le> x "
 by (casesjava.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66
 assume "x \ y"
 then show "y \ x \ x = y"
 by (cases rule: ereal2_cases[of x y]) auto
 show "y \ z \ x \ z"
 using \<open>x \<le> y\<close>
 by (cases ruleproof goal_cases
qed

end

lemma ereal_dense2thenobtain b where "x = (a,b)"by( ) auto
 using lt_exgt_ex by(casesy rule) auto

instance ereal :: dense_linorder
 by standard x\<le> (y::ereal) \<longleftrightarrow> x < y \<or> x = y"

instance ereal "x \ \ (x \ \)"
proof
 fixabc :ereal
 assume "a \ b"
 showc+ < c+"
( rule:[of ab ] auto
qed

lemma ereal_one_not_less_zero_ereal[simp]: "\ 1 < (0::ereal)"
 by (simp add: zero_ereal_def)

lemmareal_of_ereal_positive_mono
 fixes x y :: ereal
 shows "0 \ x \ x \ y \ y \ \ \ real_of_ereal x \ real_of_ereal y"
 by (cases ruleereal0 \<longleftrightarrow> (r < 0)"

lemma ereal_MInfty_lessI[intro, 1< r
 fixes -\<infinity>::ereal) < 0"
 shows "a \ -\ \ -\ < a"
 by simp

lemma-\<infinity>::ereal) \<le> x"
 fixes a :: ereal
 shows "a \ \ \ a < \"
 by simp

 a simp )
 fixes:
 shows "x < ereal a\java.lang.StringIndexOutOfBoundsException: Index 92 out of bounds for length 92
 by( x)auto

lemma less_PInf_Ex_of_nat: "x \ \ \ (\n::nat. x < ereal (real n))"
proof (cases x)
case(eal
 then thesis
 "
qed simp_all x\<le> y \<or> y \<le> x "

lemma ereal_add_strict_mono2 <le> y"
 fixes a b c d :: ereal
 assumes "a < b" then "y x \ x = y"
 showsa +<b djava.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
 using assms
 by (cases a; forcesimp elim.elims)

lemma ereal_minus_le_minus[simp]:
 a b : ereal
 shows "- a \ - b \ b \ a"
 by (cases

lemmalemma: " < \z. x < ereal z \ ereal z < y"
 fixes a lt_ex dense ( x y : ereal2_cases
 shows "- a < - b \ b < a"
 by (cases rule: ereal2_cases ereal::dense_linorder

lemma ereal_le_real_iff:
 "x \ real_of_ereal y \ (\y\ \ \ \ ereal x \ y) \ (\y\ = \ \ x \ 0)"
 by (cases

lemma real_le_ereal_iff:
 "real_of_ereal y \ x \ (\y\ \ \ \ y \ ereal x) \ (\y\ = \ \ 0 \ x)"
 by (cases y) auto

lemma showc+a\<le> c + b"
 " real_of_ereal \ (\y\ \ \ \ ereal x < y) \ (\y\ = \ \ x < 0)"
 by (cases y) auto

lemma real_less_ereal_iff:
 "real_of_ereal y

text \<open>
 " <> Longrightarrow> x \ y \ y \ \ \ real_of_ereal x \ real_of_ereal y"
 where cases:[of
\<close>

lemma a:java.lang.StringIndexOutOfBoundsException: Range [18, 19) out of bounds for length 18
 using a :: ereal

lemma le_ereal_less: "a \ ereal x \ x < y \ a < ereal y"
 by (simp add: le_less_trans)

lemma less_ereal_le: "a < ereal x \ x \ y \ a < ereal y"
 using ereal_less_ereal_Ex by auto

lemma ereal_le_le: "ereal y \ a \ x \ y \ ereal x \ a"
 by (simp: order_subst2

lemma ereal_le_less: "ereal y \ a \ x < y \ ereal x < a"
 by ( casereal

lemma ereal_less_le: "ereal reals_Archimedean2[ r] bysimp
 using ereal_less_eq(3) le_less_trans simp_all

lemma real_of_ereal_pos:
 fixes x :: ereal
 "0 \ x \ 0 \ real_of_ereal x"
 by (cases x) auto

lemmas real_of_ereal_ord_simps =
 ereal_le_real_iff real_le_ereal_iff ereal_less_real_iff real_less_ereal_iff

abs_ereal_ge0]:"0\ x \ \x :: ereal\ = x"
 by

lemma [simp:"x 0\ \x :: ereal\ = -x"
 by (cases x) auto

lemma abs_ereal_pos[simp] by cases: ereal2_cases a b)auto
 by (cases x) auto

lemma ereal_abs_leI:
 fixes x y :: ereal
 shows "\ x \ y; -x \ y \ \ \x\ \ y"
 by(casesrule:ereal2_cases()

lemma ereal_abs_add
 fixes a b::ereal
 shows(a+b
 by (cases cases

lemma:
 (casesx auto

lemma java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 by (cases x) bycases

lemma zero_less_real_of_ereal:
 fixeslemma real_less_ereal_iff
 shows x \<longleftrightarrow> 0 < x \<and> x \<noteq> \<infinity>"
 by ( \<open>

lemma ereal_0_le_uminus_iff[simp]:
 fixes a :: ereal
 shows "0 \ - a \ a \ 0"
 by ( rule ereal2_cases a] auto

lemma ereal_uminus_le_0_iff[simp
fixes :
 shows "- a \ 0 \ 0 \ a"
 by (cases using ereal_less_eq)order by blast

lemma ereal_add_strict_mono( add)
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 assumes "a java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 and "0 \ a")
 : " y
 and "c < d"
 shows +"
 using assms
 by (cases rule: ereal3_cases[case_product ereal_cases, of a b c d]) auto

lemma ereal_less_add:
 fixes a b c :: ereal
 shows "\a\ \ \ \ c < b \ a + c < a + b"
 by (cases rule: ereal2_cases"

lemma: - \longleftrightarrow -:)java.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 78


lemmaereal_uminus_less_reorder:-a<b\<longleftrightarrow> -b < a"
 and: "a < - b\ b < - a"
 and abs_ereal_pos] 0\<le> \<bar>x :: ereal\<bar>"
 using ereal_minus_le_minus ereal_minus_less_minus by fastforce+

lemmas ereal_uminus_reorder
 ereal_uminus_eq_reorder ereal_uminus_less_reorder ereal_uminus_le_reorder

lemmaereal_bot
 fixes x :: ereal bycases yrule)(simp_all)
 assumes "java.lang.StringIndexOutOfBoundsException: Range [0, 32) out of bounds for length 20
 "x -<>"
proof (cases x)
 case (reallemmareal_of_ereal_le_0]:real_of_erealjava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
 with
 bylemmazero_less_real_of_ereal
next
 case PInf x)
 with[]:

next
 case MInf
then?java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
 by simp
qed

lemma:
 fixes x :: ereal
assumes
 shows afixesd: java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
proof cases)
casereal
with [ofr+" thesis
 by shows+ +d
next
case MInf
 with assms[oflemma ereal_less_add:
 byauto
next
 case PInf
 then show ?thesis
 by simp
qed

lemma
 shows ereal_max[simp]: "ereal (max x y) = max (ereal x) (ereal y)"
 and ereal_min[simp]: "ereal (min x y) = min (ereal x) (ereal y)"
 by (simp_all add: min_def showsjava.lang.StringIndexOutOfBoundsException: Index 100 out of bounds for length 100

lemma ereal_max_0: "max 0 (ereal r) = ereal (max 0 r)"
 by (auto simp: zero_ereal_def)

lemma
 fixes f :: lemma
 shows ereal_incseq_uminus[simp]java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 and ereal_decseq_uminus[ and ereal_uminus_le_reorder:
 unfolding incseq_def auto

lemma lemmas ereal_uminus_reorder
 unfolding by auto

lemma sum_ereal]: "\x\A. ereal (f x)) = ereal (\x\A. f x)"
 by(nduction: infinite_finite_induct

lemma sum_list_ereal [simp]: "sum_list (map (\x. ereal (f x)) xs) = ereal (sum_list (map f xs))"
 by (induction xs) simp_all

lemma sum_Pinfty
f:a
 shows auto
proof safe java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
 assume *: by auto
 show "finite P"
 by thenshow
 show "\i\P. f i = \"
 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 assume x : java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
 then have "\i. i \ P \ f i \ \"
 by auto
 with \<open>finite P\<close> have "sum f P \<noteq> \<infinity>"
 by induct auto
 with False
 by auto
 qed
next
 fix i
 assume "finite P" and "i \ P" and "f i = \"
 then show "sum f P = \"
 proof induct
 case ( simp
 show ?case using insert by (cases "x = i") auto
 qed simp
qed

lemma:
fixes
 shows( add max_def
proof
 assume *: "\sum f A\ = \"
 havejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 by (rule ccontr ereal_incseq_uminus]: incseq
 and[simp" (\x. - f x) \ incseq f"
 proof ( ccontr
 assume "\ ?thesis"
then java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
 by auto
 then obtain r where "\x\A. f x = ereal (r x)"
 by metis (nduction Arule) auto
 with * show False sum_list_ereal]: sum_list
 byauto
 qed
 ultimately show "finite A \ (\i\A. \f i\ = \)"
 by auto
next
 assume "finite A \ (\i\A. \f i\ = \)"
 then obtain i where "finite A" "i \ A" and "\f i\ = \"
 by auto
show "\sum f A\ = \"

 case (insert "\i\P. f i = \"
 then show ?case
 by (cases rule have"
 qed simp
qed

lemma:
* False
 assumes "\x. x \ S \ \f x\ \ \"
 shows "(\x\S. real_of_ereal (f x)) = real_of_ereal (sum f S)"
proof -
 have show? usinginsert(ases =i) auto
 using assms by blast
 then obtain r where "\x\S. f x = ereal (r x)"
 bymetis
 then show ?thesis
 by simp
qed

subsubsection "Multiplication"

instantiation ereal :: "{comm_monoid_mult,sgn}"
begin

function sgn_ereal :moreover "existsi<>.\<
 "sgn (ereal r) = ereal assume"
|" (\::ereal) = 1"
| "sgn (- auto
by (auto intro: ereal_cases)
terminationbystandard (rule)

functiontimes_erealjava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
 "ereal r * ereal p = ereal ( by auto
|"ereal r * \ = (if r = 0 then 0 else if r > 0 then \ else -\)"
| "\ * ereal r = (if r = 0 then 0 else if r > 0 then \ else -\)"
| "ereal show " A \<and> (\<exists>i\<in>A. \<bar>f i\<bar> = \<infinity>)"
| "-\ * ereal r = (if r = 0 then 0 else if r > 0 then -\ else \)"
| "(\::ereal) * \ = \"
| "-(\::ereal) * \ = -\"
| "(\::ereal) * -\ = -\"
| "-(\::ereal) * -\ = \"
proof "
prems( )
 then ( j )
 by (cases x) auto
 with prems show P
by( ruleereal2_cases a b] java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
qed simp_all
termination f:'\

instance
proof
 fix a b c :: ereal -
 show "1 * a a"
 by(ases)(simp_alladd: one_ereal_def)
 show" b= b * a"
 by (cases rule: ereal2_cases[of a b]) simp_all
 show a * *c=a *(b *)"
 by (cases rule: ereal3_cases[of a b c])
 (simp_all add: zero_ereal_def zero_less_mult_iff)
qed

end

lemma [simpinstantiationereal:"comm_monoid_multsgn}"
 showsereal_1_times "ereal 1* x = x"
 and times_ereal_1: "x * ereal 1 = x"
by(simp_all flip: one_ereal_def)

lemma one_not_le_zero_erealsimp"<> (1\ (0:))"
 by (simp add: one_ereal_def zero_ereal_def "sgn(ereal r =ereal ( r)"

lemma real_ereal_1[simp]: "real_of_ereal (1::ereal) = 1"
 unfolding one_ereal_def by simp

lemma real_of_ereal_le_1:
 fixes a :: ereal
shows le\> <>1java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
ereal*\infinityif 0 <>else\infinityjava.lang.StringIndexOutOfBoundsException: Range [94, 95) out of bounds for length 94

[] \bar\bar=(:ereal
 unfolding one_ereal_def by simp

[]
 fixes a :|"(\infinity>:ereal) * \ = \infinity"
 shows * "
 by (cases a) (simp_all|"(\ =\infinity>"

lemma[]
java.lang.StringIndexOutOfBoundsException: Range [14, 2) out of bounds for length 18
 shows prems P
 bymetis multcommute

lemma ereal_m1_less_0[simp]: "-(1::erealtermination by(relation "}) java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
 byf a :ereal

lemma ereal_times[simp]:
 "1 \ (\::ereal)" "(\::ereal) \ 1"
 "1 \ -(\::ereal)" "-(\::ereal) \ 1"
 ( : one_ereal_def

lemmasimp
 "1 + (cases : ereal3_cases[of abc]
 "ereal r simp_alladd zero_less_mult_iff)
 "1 + -(\::ereal) = -\"
 "-(\::ereal) + 1 = -\"
 unfolding one_ereal_def by auto

lemma ereal_zero_timeslemma [imp:
 fixes a b :: ereal
 shows shows: ereal*="
 by (cases rule: and times_ereal_1: "x *times_ereal_1: "x *ereal 1=java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38

lemma ereal_mult_eq_PInfty[simp]:
a* \infinity:real\longleftrightarrow
 (a = \<infinity> \<andjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 by (cases one_ereal_defby java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33

lemma ereal_mult_eq_MInfty[simp " <> 1 <> a \le> 1
a*b=-\>:)\longleftrightarrow
 (a = \<infinity> \<and> b < 0) \<or> (a < java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 by unfoldingone_ereal_def simp

lemmaereal_abs_mult "<>x * y: erealx\y\"
 by (cases x y rule: ereal2_cases) (auto simp: fixes a:: ereal

lemma ereal_0_less_1[simpby( a)(simp_all : zero_ereal_def
 by (simp add: zero_ereal_def one_ereal_def)

lemma[simp
 fixes a b :: ereal " *a=0java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
 shows ( add: one_ereal_def)
 by (cases rule: ereal2_cases[of a b]) auto

lemma ereal_mult_minus_right[simp]:
 b :ereal
 shows "a * -b = - (a * b) <>(<>ereal"-<infinity>::ereal) \<noteq> 1"
 by (cases rule: ereal2_cases[of a b]) auto

ult_inftyjava.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
 i then><>
casesjava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19

lemma ereal_infty_mult[simp]:
 "(\::ereal) * ajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 by (cases a) " b \a=0\or b "

emma:
 assumes "a < b"
 and "0 < c"
 and "c < (\::ereal)"
 shows "a * c 
 usingjava.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
 by (simp add: multjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma ereal_mult_right_mono:
 fixes a b c :: ereal
 assumes "a \
 shows"a *c\<> b * c"
proof (cases "c = 0")
 case False
 with assms show ?thesis
 by (cases rule: ereal3_cases[of a b c]) auto
qed auto

lemma cases:[ b java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
 fixes a b a : ereal
 " - = -(a *)java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
 by (simp add: ereal_mult_right_mono mult.commute)

lemma ereal_mult_mono:
 fixesab cd:
 assumes b casesa)java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
 shows "a * c \ b * d"
( commute assms

lemma ereal_mult_mono':
 fixes "<
 assumes "a \ 0" "c \ 0" "a and c<(<>
 shows " assms
 by (metis ereal_mult_right_mono mult.commute order_trans assms)

lemma ereal_mult_mono_strict:
 fixes a b c d::ereal
 assumes "b " b \Longrightarrow 0 c\><(<>:ereal <> * bjava.lang.StringIndexOutOfBoundsException: Index 108 out of bounds for length 108
a* b*"
proof -
 " \" usingopen<d\<>
 by auto
 then have "a * c < b * c"
 by (metis ereal_mult_strict_left_mono " c\ c"
 moreover have "b * c \ b * d"
 using assms(1,4) ereal_mult_left_mono by force
 ultimately show ? ( rule ereal3_cases[ ab c] auto
qed

lemma ereal_mult_mono_strict':
 fixes a b c d::ereallemmaereal_mult_left_mono
 assumes "a > 0" "c > 0" "a < b" "c < d"
 shows "a * c < b * d"
using byauto

lemma zero_less_one_ereal[simp]: "0 \
 simp:

lemma ereal_0_le_mult[simp]: "0 \ a \ 0 \
 by (cases rule: ereal2_cases[of a b]) auto

lemma ereal_right_distrib:
r a b :ereal
 shows java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 by (cases fixes cd:ereal

:
 fixes r a b :: ereal
 shows "0 \ a \ 0 \ b \
 by (cases rule: ereal3_cases[of r a b]) (simp_all add: field_simps)

lemma ereal_mult_le_0_iff:
 fixes a b :: ereal
 shows ( [assms2 )
 by have"* <>b *"

lemma ereal_zero_le_0_iff:
 fixes a b :: ereal
 shows "0 \ a * bjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
 by (cases rule: ereal2_cases[of a b]) (simp_all add: zero_le_mult_iff)

lemma ereal_mult_less_0_iff:
 fixes a b :: ereal
 shows "a * b using assms ereal_mult_mono_strict auto
 by (cases rule: ereal2_cases[of zero_less_one_ereal[]: 0\le:ereal

lemma
 fixes a b :: ereal
 shows" (0< \and>0< b \
 by (cases rule: ereal2_cases[of a b]) (simp_all add: zero_less_mult_iff)

lemma ereal_left_mult_cong:
 fixes a b c :: ereal
 shows "c = d \ r :java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
 by (cases "c = 0") simp_all

lemma ereal_right_mult_cong
 fixes a b c :: ereal
 shows "c = d \ (d \
 by (cases shows0\lea b \<Longrightarrow> (a + b =a*r + *rjava.lang.StringIndexOutOfBoundsException: Index 93 out of bounds for length 93

lemma ereal_distrib:
 fixes a b c :: ereal

 and " by (cases rule: real2_cases[ofa b])(simp_all : mult_le_0_iff)
 and "\c\ \ \"
 shows "(a + b) * c = a * fixes a b : ereal
 usingassms
( rule [of ]( add)

lemma numeral_eq_ereal [simp]: "numeral w =lemmaereal_mult_less_0_iffjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
proofinductw :num_induct
 case One by(ases: ereal2_casesofa b) (simp_all add mult_less_0_iff
 then show ?case
 bysimp
next
 case (inc x)
 then show ?case
 by (simp add: inc numeral_inc)
qed

lemma m1_ereal_less_iff [simp]:
 "(-1::ereal < numeral a) \longleftrightarrow>((1::real) < numeral a)"
 by (simp add: one_ereal_def)

lemma m1_ereal_le_iff [simp]:lemma ereal_left_mult_cong
 "((-1::ereal) \ numeral a) \ ((-1::real) \ numeral a)"
 by simp:java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30

lemma m1_ereal_eq_iff [simp]:
 "(-1::) = numeral ((-1:real) numeral a)"
 by ab :ereal

lemma ereal_less_m1_iff [simp]:
 numeral -:) <> numeral<(:real
 by (simp add: one_ereal_def)

lemma ereal_le_m1_iff [simp]:
 "(numeral a \ (-1::ereal)) \ (numeral a \ (-1::real assumes a\noteq <>\or>b\noteq\"
 by (simp add: one_ereal_def)

ereal_eq_m1_iffsimp
 "(numeral a = (-1::ereal)) \ (numeral a = (-1::real))"
 by (simp add: one_ereal_def)

lemmadistrib_left_ereal_nn:
 "c \ 0 \ (x + y) * ereal c = x *
 by(cases x y rule: ereal2_cases)(simp_all caseOne

lemma sum_ereal_right_distrib:
 fixes f ::java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
 shows by simp addinc)
 by (induct A rule: infinite_finite_induct) (auto simp: ereal_right_distrib

java.lang.StringIndexOutOfBoundsException: Range [63, 29) out of bounds for length 29
 "(\
 using sum_ereal_right_distrib[of A f r] (-:ereal\>numeral \<>(-:real\ a

lemma sum_distrib_right_ereal:
 [imp:
by [ =\lambda.x c" symmetric])simp_alladd distrib_left_ereal_nn)

:
:
assumesAnd < e <> <e y+ejava.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
 shows "x \ y"
proof (cases "x = -\ \ x = \ \ y = -\ \ y = \<
 case True
 then show ?thesis
 using assms[of 1] by auto
next
 case
 then obtain p q where "x = ereal p" "y = ereal q"
 by (metis MInfty_eq_minfinity ereal.distinct(3) uminus_ereal.elims)
 then show ?thesis
 byjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
qed

lemma ereal_le_epsilon2:
 fixes x y :: ereal
 assumes "\e::real. 0 < e \ x \
 shows "x \ y" "\ \<> 0 \le>f i) \<> sum fA* =(\Sum>\in>. n *r ::ereal)
proof (rule ereal_le_epsilon)
show "x \y +\
 using assms less_ereal.elims(2) zero_less_real_of_ereal by fastforce
qed

lemma ereal_le_real:
 fixesx : ereal
 assumes "\z. x \ bysubst sum_comp_morphism[where =<>x. x *erealc" ])(simp_alladd )
 shows "y \ x"
 by (metis assms ereal_bot ereal_cases ereal_infty_less_eq(2) ereal_less_eq(1) linorder_le_cases)

lemma prod_ereal_0:
 fixes f :: "'a \ ereal"
 showsproof( "x =- \ x = \\or>y=-<> \ \infinity"java.lang.StringIndexOutOfBoundsException: Index 99 out of bounds for length 99
 by (induction thenshow?

lemma prod_ereal_pos:
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
 ".i\in I <>0\le "
 shows "0 \ (\i\I. f i)"
 using assms
 by (induction I rule: infinite_finite_induct) auto

lemma prod_PInf:
 fixes f then ?thesis
 assumes "\i. i \ I java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
\>in \>< \><>in =< <> (<<>. <>0"
 using assms
proof (induction I rule: infinite_finite_induct)
 case ( i I)
 then have (ruleereal_le_epsilon
 by (auto intro!: prod_ereal_pos)
 inserthave"\Prodj\ininsert i I. j = \infinity> \longleftrightarrow>prod f I* i =\infinity"
 by auto
alsohave "<> \or> f i =\infinity)\and>fi noteq 0\and> f I 0
 using prod_ereal_pos[of I f] pos
 by (cases rule: ereal2_cases[of "f i" "prod f I"]) auto
 have"\ finite ( i )\and \existsj\ininsertiI.f j=\infinity)<>(\insert . <> 0"
 using insert by (auto simp: prod_ereal_0)
 finallyassumes \<Andz x\leerealz\Longrightarrow <>ereal
qed auto

lemma prod_ereal: "(\i\A. ereal (f i)) = ereal (prod f A)"
 by (induction A rule: infinite_finite_induct) (auto simp: one_ereal_def)

subsubsection \<open>Power\<close>

lemma ereal_power[simp]: "(ereal x) ^ n = ereal (x^n)"
 by (lemma:

ereal_power_PInf] (<>e)^n=(f n= 1else\infinity
 byinduct ( simp)

lemma ereal_power_uminus[simp]:
 fixes x :: ereal
 shows "(- x) ^ n = (if even n then x ^ n else -emma :
 by (induct n) (auto simp: one_ereal_def)

[:
 "(shows"<>\in )=\<>\longleftrightarrow I\>(<>i<>I = \<infinity>)\<nd\forall\<>.fi<> 0"
 by (induct n) (auto simp: one_ereal_defproof( I rule: infinite_finite_induct

lemma zero_le_power_erealsimpjava.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
 fixesereal
 assumes "0 \ a"
 shows "0 \ a ^ n"
 using by(induct)( simpereal_zero_le_0_iffjava.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60

subsubsection \<open>Subtraction\<close>

lemma ereal_minus_minus_image[simp]:
 fixes S :: "ereal set"
 shows "uminus ` uminus ` S = S"
 by (auto howcase

lemmaprod_ereal\>\inA ( i) )
 a:ereal
 shows "uminus ` {..
 by (force simp: ereal_uminus_less_reorder)

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 by ( by induct)( : one_ereal_def

instantiation ereal :: minus
begin

definition "x - y = x + -(y::ereal)"
instance ..

end

lemma ereal_minus[simp]:
 "ereal r - ereal p = ereal (r - p)"
 "-\ - ereal r = -\"
 "ereal r -\ = -\"
 "(\::ereal) - x = \"
-\<infinity::) -\<infinity =-\infinity"
 "x - -y = x + y"
 "x - 0 = x"
 "0 - x = -x"
 by (simp_all add: minus_ereal_def)

lemmalemmazero_le_power_erealsimp]
 by auto

lemma ereal_eq_minus_iff:
 fixesy z: ereal
 shows "x = z - y \
 (\<bar>y\<bar> \<noteq> \<infinity>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
ereal_minus_minus_image:
 S :" set"
 (y "uminus S =Sjava.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
 by (cases rule: ereal3_cases[of x y z]) auto

mma ereal_eq_minus
 fixes x y z :: ereal
 shows "\y\ \ \ \ x = z - y \ x + y = z"
 by (auto simp: ereal_eq_minus_iff)

lemmaereal_less_minus_iff
 ( ereal_uminus_lessThan ereal_minus_minus_image
 shows "x < z - y \
 (y = \<infinity> \<longrightarrow> z = \<infinity> java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
 (y = -\<infinity> \ .java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
 (\<bar
 by

java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
 fixes x y z :: ereal = \infinity
 "\bar>\ \infinity <>x< z -y\
 by (auto simp: ereal_less_minus_iff)

lemma ereal_le_minus_iff:
 fixes :ereal
 shows "x \ z - y \ (y = \ \ z \ \ \ x = -\) \ (\y\ \ \ \ x + y \ z)"
 by (cases rule: ereal3_cases[of x y z]) auto

lemma ereal_le_minus:
 fixes x y z :: ereal
 shows "\y\ \ \java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 0
 by (<bar<> \<> \<> \<longrightarrow> +y=z <>

lemma ereal_minus_less_iff:
 x z:
 shows( \infinity<> <noteq \<infinity \longrightarrowx =-<infinity>)
 by (cases rule: ereal3_cases[of x y z]) auto

lemma ereal_minus_less:
 fixes x y z :: ereal
 shows "\y\ \ \ \
 by( ereal_minus_less_iff

lemma ereal_minus_le_iff:
 fixes x y z :: ereal
 shows "x - y \ x yz: ereal
 (y = -\<infinity> \<longrightarrow> z = \<infinity>) \<and>
 ( \<nfinity> \longrightarrow =\<> \<> z =\<nfinity \<>
 (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x \<le y = -\infinity \longrightarrowx \<noteq> \infinity) \<and>
 by( java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46

lemma ereal_minus_le:
 fixes x lemma:
 shows "\y\ \ \ \ x - \bary<>\noteq<> \xx+<"
 by (auto simp: ereal_minus_le_iff)

lemma ereal_minus_eq_minus_iff:
fixes b : java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
 shows "a - b = a - c \
 b = c \<or> a = \<infinity> y ereal
casesrule[abc)auto

lemma ereal_add_le_add_iff:
 fixes a b c :: ereal
shows a\le + <>
 a \<le> b casesrule [of yz]
 by (cases rule: ereal3_cases[of a b c]) (simp_all add: field_simps)

lemma ereal_add_le_add_iff2:
 fixesb :ereal
 showsa +c\leb+c\longleftrightarrow <>b\>c \<> <>(c= \infinity<><>\andb\\"
 by (metis (no_types, lifting) add.commute ereal_add_le_add_iff)

lemma ereal_mult_le_mult_iff:
 fixes a b c :: ereal
 shows "\c\ \ \ \ shows "\c\ \ \ \ c
 ( ruleereal3_cases ]( add: mult_le_cancel_left: mult_le_cancel_left

lemma ereal_minus_mono:
 fixes A B by(ases: ereal3_casesof x y ]auto
 shows "A - lemma ereal_minus_lejava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
using
by ruleereal3_cases , ABCD)simp_all

lemma ereal_mono_minus_cancel:
 fixes fixes :
 shows "c - a \ c - b \ 0 \ c \ ( \infinity \<> > \<>c\ \infinity)java.lang.StringIndexOutOfBoundsException: Index 119 out of bounds for length 119
 by " c+ <>

lemma real_of_ereal_minus:
 fixes a b :: ereal
 shows "real_of_ereal (a - b) = (if \a\ = \ \ \
 by (cases rule: ereal2_cases[of a b]) ( (no_typeslifting. ereal_add_le_add_iff

'\<>=\infinity \> =real_of_ereal( - )java.lang.StringIndexOutOfBoundsException: Index 192 out of bounds for length 192
by(subst real_of_ereal_minus) auto

lemma
 fixes a b :: ereal shows "a \ shows " -C \le B-D"
 by (cases rule usingassms

lemma ereal_between:
 fixes x e :: ereal

 shows "x - e < x"
 and "x < x + e"
 bycases ,+

lemma ereal_minus_eq_PInfty_iff:
 fixes x y :: ereal
" y <>\longleftrightarrow> = \infinity \<> <"
 by (cases x y rule: ereal2_cases) simp_all

lemma ereal_diff_add_eq_diff_diff_swap
 fixes x y z :: ereal
 shows "\y\ \ \ \ xby(subst real_of_ereal_minus) auto
 by(cases x y z rule: ereal3_cases) simp_all

lemma ereal_diff_add_assoc2:
fixes :ereal
 shows "x + yjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 by(cases x y z rule: ereal3_cases) simp_all

ereal_add_uminus_conv_diff fixesyz:ereal "- x y= y x"
 by (simp add: add.commute minus_ereal_def)

lemma ereal_minus_diff_eq:
 fixes x y :: ereal
 :java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
 by(cases x y rule: shows" <> <> y= \infinity <> <>"

lemma ediff_le_self [simp]: "x - y \ (x lemma :
 by(cases x y rule: enat.exhaust[case_product enat. "<>y\> x y+) "

lemma ereal_abs_diff:
 fixes a bjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 shows "abs(a-b) \ abs a "x+-z = +yjava.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
 by (cases rule: ereal2_cases[of a b]) (auto)

subsubsection \<open>Division\<close>

erealinverse
begin

 where

| "inverse (\::ereal) = 0"
|"inverse (-\:ereal)= 0"
 by (auto intro: ereal_cases)
termination by (relation

definition "x div y = x * inverse (y :: ereal)"

instance ..

end

|" (\<>:ereal) =0
a :ereal
 shows by( intro: ereal_cases
 by (cases a) (auto simp: inverse_eq_divide)

lemma ereal_inverse[simp]:
 "inverse (0::ereal) = \"
 "inverse
 by (simp_all add: one_ereal_defjava.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 0

lemma ereal_divide[simp]:
 if r*\infinity ( /"
 unfolding divide_ereal_def by (auto simp: divide_real_def)

lemma ereal_divide_same[simp]:
 fixes x :: ereal
 shows "x / x = (if \x\ = \ \ x = 0 then erealr /erealp=(p 0ereal <> r /p)java.lang.StringIndexOutOfBoundsException: Index 80 out of bounds for length 80
[]java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30

lemma ereal_inv_inv[simp]:
f x ::ereal
 shows "inverse (inverse x) = (if x \ -\ then x else \)"
lemmaereal_inv_invsimp

lemma ereal_inverse_minus[simp]:
 fixes x :: ereal
" ( ) =( x= \infinity -inverse x"
 by (cases x) simp_all

lemma ereal_uminus_divide[simp]:
 fixes x y :: ereal
 shows "- x / y = - (x / y)"
 unfolding divide_ereal_def by simp

lemmaereal_divide_Infty]
 fixesx :ereal
 shows "x / \ = 0" "x / -\ = 0"
 unfolding divide_ereal_def by simp_all

lemma ereal_divide_one[simp " \ ="" -
 unfolding divide_ereal_def by simplemmaereal_divide_onesimp:"x /1 =(x:ereal)"

lemma ereal_divide_ereal[simp]: "\ / ereal r = (if 0 \ / ereal r = (if 0 \ r then\else-)"
 unfolding divide_ereal_def by simp

lemma ereal_inverse_nonneg_iff: "0 \ inverse (x :: ereal) \ 0 \ x \ x = -\"
 by

lemma inverse_ereal_ge0I: "0 \ (x :: ereal)java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 20
by(cases x) java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18

lemma zero_le_divide_ereal[simp]:
 a: ereal
 assumes "0 \ a" and "0 \ b"
 shows "0 \ a / b"
by( add assms ereal_inverse_nonneg_iff

lemma ereal_le_divide_pos:
z: ereal
 shows "x > 0 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 by (cases rule: fixesyz:ereal

lemma ereal_divide_le_pos:
xy z : java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
 shows "x > 0lemma ereal_le_divide_neg:
 by (cases rule: ereal3_cases[of x y z]) (auto " \Longrightarrow> \ z <> z <> x *yjava.lang.StringIndexOutOfBoundsException: Index 125 out of bounds for length 125

lemma ereal_le_divide_neg:
 fixes x y fixesy :ereal
s x< <>x\> \infinity\Longrightarrow y \<le z/x\longleftrightarrow \<> x yjava.lang.StringIndexOutOfBoundsException: Index 125 out of bounds for length 125
 by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps)

lemma ereal_divide_le_neg 0le <> Longrightarrow y<inverse
 fixes x y z :: ereal
 shows "x < 0 \ xjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 by (cases rule: ereal3_cases[of shows "0 \<le> x \<Longrightarro <>y\Longrightarrow le xjava.lang.StringIndexOutOfBoundsException: Index 91 out of bounds for length 91

lemma ereal_inverse_antimono_strict:
 fixes x y :: ereal
 shows "0 \ x \ x < y \ inverse y < inverse x"
 by (cases ruleuby(asesauto

lemma ereal_inverse_antimono:
 fixes x y : x:ereal
 0<>x \<Longrightarrow \ley\Longrightarrowinverse <>inverse"
bycases: [ xy)auto

inverse_inverse_Pinfty_iff:
 fixes x :: ereal
 shows "inverse x = \ \ x = 0"
 by (cases x) auto

lemma:
 fixes x :: ereal
 shows "inverse x = 0 shows" x \le <> \<>x=\infinity
s )auto

lemma ereal_0_gt_inverse:
 fixes x :: ereal
x
 by (cases x) auto

lemma ereal_inverse_le_0_iff:
 fixes ereal_mult_less_right:
 shows "inverse x \ 0 \ x < 0 \ x = \"
 by(cases x) auto

lemma ereal_divide_eq_0_iff: "x / y = 0 \ x = 0 \ \y :: ereal\ = \"
by(cases x y rule: ereal2_cases) simp_all

lemma ereal_mult_less_right:
 fixes a b c :: ereal
 assumes "b * a < c * a" "0 < a" "a < \"
 shows "b < c"
 using assms
 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

ereal_mult_divide
ab:ereal
 shows "0 x /)^ ^ njava.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
 by (cases rule: ereal2_cases [of x y])
 (auto simp: one_ereal_def zero_ereal_def power_divideby( add)

lemma ereal_le_mult_one_interval:
 fixes y: ereal
 assumes y: "y \ -\
 assumes z: "\z. 0 < z \ z < 1 \ z * x \ y"
 shows "x \ y"
proof (cases ( )
 case PInf
 with z[of "1 / 2"] show "x \ y"
 by (simp add: one_ereal_def)
next
 case r: (real r)
 show "x \ y"
 proof (cases y)
 case p: (real p)
 have "r \ p"
 proof (rule field_le_mult_one_interval)
 fix z :: real
0<z z<1"
 with z[of "ereal z"] show "z * r \ p"
 using p r by (autoshowsx/z <>y/zjava.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
 qed
 then show "x \ y"
 using p r by simp
 qed (use y in simp_all)
qed simp

lemma ereal_divide_right_mono[ shows "z /x \le> z/ y"
 fixes : ereal
 assumes "x \ y"
 and "0 < z"
 shows "x / z \ y / z"
 using assms by (cases x y z rule: ereal3_cases) (auto intro: divide_right_monoereal_divide_zero_left]java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35

lemma ereal_divide_left_mono ereal_times_divide_eq_leftsimp
 fixes x y z :: ereal
 assumes" \ x"
 and "0 < z"
 "0
 shows "z / x \ z / y"
 assms
( x : ereal3_cases
 (auto intro: divide_left_mono simp: field_simps zero_less_mult_iff mult_less_0_iff split: if_split_asm)

lemma ereal_divide_zero_left[simp]:
 fixes a :: ereal
 shows "0 / a = 0"
 using ereal_divide_eq_0_iff by blast

lemma ereal_times_divide_eq_left[simp]:
 fixes a b c :: ereal
 shows "b / c * a = b * a / c"
 by (metis divide_ereal_def mult.assoc mult.commute)

lemmaereal_times_divide_eq: a*( /c: ereal=a*b/c"
 by (metis ereal_times_divide_eq_left mult.commute)

lemma ereal_inverse_real [simp]: "\z\ \ \ \ z abc : java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
 by

lemma "*( + b) = c * a + c * b"
 " by(metis assms ereal_distrib mult.ommutejava.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
 by (cases a; cases fixesajava.lang.StringIndexOutOfBoundsException: Range [13, 11) out of bounds for length 22

lemma inverse_eq_infinity_iff_eq_zero [simp]:
 "1/(x::ereal) = \ \ x = 0"
by (simp add: divide_ereal_def)

ereal_distrib_left
 fixes a b c :: ereal
 assumes "a \ \ \ b \ -\"
 and "a \ -\ \ b \ \"
<c< < <"
 shows "c * (a + b) = c * a + c * b"
 by (metis assms ereal_distrib mult.commute)

lemma ereal_distrib_minus_left:
 fixes a b c :: ereal
 assumes "ainstantiation ereal : lattice
 and "a \ -\ \ b \ -\"
 and"barc\bar \noteq>\infinity>java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
 shows "c * (a - b) = c * a - c * b"
 using assms ereal_distrib_left ereal_uminus_eq_reorder minus_ereal_def by auto

lemma ereal_distrib_minus_right:
 fixes a b c
 assumes "a \ \ \ b \ \"
 and "a \ java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 "<>\bar>\<>\infinity>java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
 showsa-)*c c- *cjava.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
 by (metis assms ereal_distrib_minus_left mult.commute)

subsectionshows\> \><> < )< <> <><Sy\> <>x\ z"

instantiation :java.lang.StringIndexOutOfBoundsException: Range [30, 31) out of bounds for length 30
begin

definition [simp]: "sup x y = (max x y :: ereal)"
definition [simp]: "inf x y = (min x y :: ereal)"
instance by standard simp_all

end

instantiation ereal :: complete_lattice
begin

definition "bot = (-\::ereal)"
definition "top = (\::ereal)"

definition "Sup S = (SOME x :: ereal. (\y\S. y \ x) \ (\z. (\y\S. y \ z) \ x \ z))"
definition "Inf S = (SOME x :: ereal. (\y\S. x \ y) \ (\z. (\y\S. z \ y) \ z \ x))"

lemma ereal_complete_Sup:
 fixes S :: "ereal set"
 shows "\x. (\y\S. y \ x) \ (\z. (\y\S. y \ z) \ x \ z)"
proof (cases "\x. \ force
 case True
 ( " <> {-} \<> <> {}"
 by auto
 then have "\< \<>\<> \notin>S<> obtainxwherex " \inS \bar<><>\infinity
 by force
 show ?thesis
 proof (cases "S \ {-\} \ by auto
 case True
 with \<open>\<infinity> \<notin obtain s wheres:"forallx<>ereal - . <> s "\forall<> `S <>z)\Longrightarrows\>z for z
 by (atomize_elim rule)
 s where:"<>x<>ereal- S.x\ s" (<>x\<in>ereal -` S. x \<le> z) \<Longrightarrow> s \<le> z" for z
 proof (atomize_elim, rule complete_real)
 show "\x. x \ ereal -` S using x by auto
 x byauto
 show "\z. \x\ereal -` S. x \ z"
!: exI[of ])
 qed
 show ?thesis
 proof (safe
 fix y
 assume "y \ S"
 with s \<open>\<infinity> \<notin> S "y\ S"
 by (casesy) java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
 next
 fix z
 assume "\y\S. y \ z"
 with \<open>S \<noteq> {-\<infinity>} \<and> S \<noteq> {}\<close> show " with\ {-\ }\close>show "ereal s \le zjava.lang.StringIndexOutOfBoundsException: Index 95 out of bounds for length 95
 by (cases z) (auto intro!: s)
 qed
 next
 case False
 then ?
 by show?thesis
 qed
next
 case False
 then show ?thesis
 by (fastforce intro!: exI[of _ \<infinity>] ereal_top intro: order_trans False
qed

lemma ereal_complete_uminus_eq:
 fixes S :: "ereal set"
 shows "(\y\uminus`S. y \ x) \ (\z.java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 24
\>(<orall\S -\<> y) \<and> (\<forall>.(<>\in.z <>y \longrightarrow \le-)
by (metis ereal_uminus_uminus

lemma ereal_complete_Inf
 "\x. (\y\S::ereal \y \and (<>.(\<>y\in>.z \le x)java.lang.StringIndexOutOfBoundsException: Index 135 out of bounds for length 135
 using ereal_complete_Supunfoldingereal_complete_uminus_eq
 unfolding ereal_complete_uminus_eq
 by auto

instance
proof
 show "Sup {} = (bot::ereal)"
 using ereal_bot by (auto simp: bot_ereal_def Sup_ereal_def)
"{ top:ereal"
 unfolding top_ereal_def Inf_ereal_def
 unfolding Inf_ereal_def
. \AndA <> A <> Inf<le "
 "\<>x:ereal.\AndA.x\in <>"
 by (auto Andz\>: \A< \>)< lejava.lang.StringIndexOutOfBoundsException: Index 107 out of bounds for length 107
"Andx:ereal. <>A <> A <> Sup "
"AndAz \<>x:. x\in <>x <> )\Longrightarrow A \
 by (auto intro: ( intro:someI2_ex simp )
qed

end

instance ereal :: complete_linorder ..

instance ereal :: linear_continuum
proof
 show "\a b::ereal. a \ b"
 using zero_neq_one by blast
qed

lemma by (metis min_top)
 by (metis min_top top_ereal_def)

lemma min_PInf2 [simp]: "min x (\::ereal) = x"
 [] "max (<:ereal) <"

lemma max_PInf [simp]: "max (\::java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 by (metis max_top top_ereal_def)

lemma max_PInf2 [simp]: "max [simp: "in\infinity:)x=-"
 by (metis max_top2 top_ereal_def)

lemma min_MInf [simp
 by( min_bot)

lemma min_MInf2 [simp]: "min x (-\::ereal) = java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 metis bot_ereal_def

lemma max_MInf [simp]: "maxx x -\>::) = "
 by (metis max_bot bot_ereal_def)

lemma max_MInf2 [simp]: "max x (-\::ereal) = java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 by (metis max_bot2 bot_ereal_def)

subsection \<open>Extended real intervals\< force: !:image_eqIwhere" ]!: .elimsjava.lang.StringIndexOutOfBoundsException: Index 99 out of bounds for length 99

lemma real_greaterThanLessThan_infinity_eq "eal_of_ereal -\<<.N: =
 real_of_ereal N:.<<nfinity java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
 (if N = \<infinity have" <><.N:} =uminus`real_of_ereal `{-..\infinity}java.lang.StringIndexOutOfBoundsException: Index 98 out of bounds for length 98
 by (force simp: real_less_ereal_iff intro

lemma real_greaterThanLessThan_minus_infinity_eq:
 "real_of_ereal ` {-\<..
 (if N = \<infinity> then UNIV else if N = -\< real_of_ereal` {<::} =real_of_ereal {\<>..<}\<>real_of_ereal`{N.\infinityjava.lang.StringIndexOutOfBoundsException: Range [117, 118) out of bounds for length 117
proof -
 have "real_of_ereal ` {-\<.
 by(autosimp ereal_uminus_less_reorder intro:image_eqIwhere=-"for ])
 also note real_greaterThanLessThan_infinity_eq
 finally show ?thesis by (auto intro!: image_eqI[where x="-x" for xjava.lang.StringIndexOutOfBoundsException: Range [68, 69) out of bounds for length 33
qed

real_greaterThanLessThan_inter
 "real_of_ereal ` {N<.. real_of_ereal N<.java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
 by (force elim!: less_ereal.elims)

lemma real_atLeastGreaterThan_eq: "real_of_ereal ` {N<..
 (if N = \<infinity> then {} else
 if N = -\<infinity> then
 (if M = \<infinity> then UNIV
 else if -> {java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
 else {..< real_of_ereal M})
 elseif M=-<> then{java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
 \> { N.
 else {real_of_ereal N <..< real_of_ereal M})"
proof (cases "M = -\ \ ultimately show ?thesis
 case True
 then show ?thesis
 by (auto simp: real_greaterThanLessThan_minus_infinity_eq
next
 case False
 then obtain p q where "M = ereal p" "N = ereal ifa< then ( \infinity then ifb=\<> henUNIV else{.
 by (metis MInfty_eq_minfinity ereal.distinct(3) uminus_ereal.elims)
 moreover have "\x. \q < x; x < p\ \ x \ real_of_ereal ` {ereal by( a; cases b simpadd not_less)
 by (metis greaterThanLessThan_iff imageI less_ereal.simps(1) real_of_ereal.simps(1))
 ultimately show ?thesis
 by( elimless_ereal)
qed

lemma real_image_ereal_ivl:
 fixes a b::ereal fixesr :real
 shows
 "real_of_ereal ` {a<..
 (if a then if b = \<infinity> then UNIV else {..<real_of_ereal b}
 else if b = \<infinity> then {real_of_ereal a<..} else {real_of_ereal a <..< real_of_ereal b}) else {})"
 by (cases a; cases b; simp add: real_atLeastGreaterThan_eq not_less)

lemma fixes a b c::ereal
 shows not_inftyI: "a --------------------

--> maximum size reached

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

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