(* 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> usingby ( add_left_mono namedfrom\<^theory>\<open>Complex_Main\<close>. thenshowshows"\.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"thenshow ?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] proofusing 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
lemma open_enat: "open {enat n}" proof(ases case 0 thenhave"{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) thenshow ?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" thenhave"A = (\n\{n. enat n \ A}. {enat n})"
(imp: set_eq_iff not_enat_eq moreoverhave"open \" byautoopen_enat ultimatelyshow"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 moreoverhave"open \" by ( open_Unopen_UN ballI open_greaterThan ultimatelyshow ?java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19 by simp next
sume " A""\ \ A" thenhave" (range lessThan \ range greaterThan) A" "\ \ A" unfolding open_enat_def by auto thenshow"\n::nat. {n <..} \ A" proofinduction case( A B) thenobtain 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) thenobtainkwhere\<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 thenhave [ 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)" byshow" \ \ (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
funwhere "- ( 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) andthenshow"(\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[simpForlemmas about extended real see by (simp_all add: infinity_ereal_def\<close>
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 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) thenobtain 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
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
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" thenshow"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
lemmashowc+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
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+
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 byshows+ +d next
case MInf with assms[oflemma ereal_less_add: byauto next case PInf thenshow ?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 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 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" bythenshow 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 thenhave"\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 = \" thenshow"sum f P = \" proof induct case ( simp show ?caseusing 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 thenobtain r where"\x\A. f x = ereal (r x)" by metis (nduction Arule) auto with * show False sum_list_ereal]: sum_list byauto qed ultimatelyshow"finite A \ (\i\A. \f i\ = \)" by auto next assume"finite A \ (\i\A. \f i\ = \)" thenobtain i where"finite A""i \ A" and "\f i\ = \" by auto show"\sum f A\ = \"
case (insert "\i\P. f i = \" thenshow ?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 - haveshow? usinginsert(ases =i) auto using assms by blast thenobtain r where"\x\S. f x = ereal (r x)" bymetis thenshow ?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_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 < b * c"
using assms by (cases rule: ereal3_cases[of a b c]) (auto simp: zero_le_mult_iff)
lemma ereal_mult_strict_left_mono: "a < b \ 0 < c " b =-\infinityereal<>
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
lemmacases:[ 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\<> byauto 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_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] byauto
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
&;open>Subtraction\<close>
lemma[simp]:
xes: ereal"
shows"uminus`uminus`S " by (auto simp: image_iff)
lemma ereal_minus "ereal r - ereal pus: "-\ - ereal r = -\"
>=-<>"
shows<><>\noteq<>\Longrightarrow z <>x+y " "-(\::ereal) -\ = -\" "x - -y = x + y" "x - 0 = x" "0 - x yz : java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22 by (simp_all add: minus_ereal_def)
lemma ereal_x_minus_x[simp]: "x - x =java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 byauto
lemma ereal_eq_minus_iff:
fixes x y z :: ereal
shows "x = z - y \
(\<>y\barnoteqinfinitylongrightarrow>x )\and
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
(y = \<infinity> \<longrightarrow> fixesy : ereal
y=<> \<longrightarrow z\>><> \infinity" by (cases rule: ereal3_cases[of x y z]) auto
lemma ereal_less_minus_iff:
fixes :ereal
shows "x < z - y \
(y = \<infinity> \<longrightarrow> z = \<infinity> \<and> xy=<nfinity <> x \infinitylongrightarrow=<>)\and
( \><> x\<\> and
(\<bar>y\<bar> \<noteq> \<infinity>\<longrightarrow> casesrule: ereal3_cases[of x y z]) auto by (cases rule: ereal3_cases[of x y z]) auto
ereal_less_minus
fixes x y z :: ereal
shows"<>\bar> <> \ by (auto simp: ereal_less_minus_iff)
lemma ereal_le_minus_iff:
fixes x y z :: ereal
shows "x \ z - y \ (y = \ \ z \ \ \ x a b c :: ereal by (cases rule: ereal3_cases[of x y z]) auto
lemma ereal_le_minus:
fixesx y z:: ereal
shows "\y\ \ \ \ x \ by( rule: ereal3_cases[of a b c] auto by (auto simp: ereal_le_minus_iff)
lemma ereal_minus_less_iff:
fixes x y z :: ereal
shows "x - y < z \ y \ -\ \ (y = \ \ x \ \ \ "c+a <> c b\longleftrightarrow by( rule:ereal3_cases xy ] auto
lemma ereal_minus_less:
fixes x y z :: ereal
shows "\y\ \ \ \ x - y fixes a bc ::ereal by"a+ c \ a \ \ a \ \infinity> \)java.lang.StringIndexOutOfBoundsException: Index 166 out of bounds for length 166
lemma ereal_minus_le_iff:
fixes x y z :: ereal
shows "x - y \ z \
(y = \<infinity bycases: ereal3_cases[ofabc]) simp_alladd)
(\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x \<le> z + y)"
( rule[ofx yz]
lemma:
fixes x y z :: ereal
shows "\y\ \ \ \ x - y \ z using assms by (auto simp: (cases: [case_productereal_cases of ]
lemma ereal_minus_eq_minus_iff:
abc: ereal
shows "a - b = a - c \
b c <>a=\infinity\ora=-<> \and b \<noteq> -\<infinityand <>-<>" by (cases rule: ereal3_cases[of a b c]) auto
lemma ereal_add_le_add_iff:
fixes a b c :: ereal
showsc +a\< +b\longleftrightarrow
a \<le> b \<or> c = \<infinity> \<or> (cjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 by (cases rule: ereal3_cases[of a b c]) (simp_all add: field_simps)
lemma ereal_add_le_add_iff2:
fixes a b c :: ereal
shows "a + c \ b + c \ a \ b \ c = \ \ (c = -\ \ a \ \ \ b \ \)" bymetis, ) addcommute)
lemma ereal_mult_le_mult_iff:
fixes a b c :: ereal
shows "\c\ \ \ \ c * a \ c * lemma real_of_ereal_minus: "\bar>\<>= \longleftrightarrow>\bary\ \Longrightarrow real_of_erealx-real_of_erealy= real_of_ereal ( - y::ereal" by (cases rule: ereal3_cases[of a b c]) (simp_all add: mult_le_cancel_left)
lemma ereal_minus_mono:
fixes A B C D :: ereal assumes "A \ "-\le> "
java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13 by (cases rule: ereal3_cases[case_product ereal_cases, of A B C D]) simp_all
lemma ereal_mono_minus_cancel:
fixes a b c :: ereal
shows "c - a \ c - b \ 0usingassms ( x,casese auto)java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43 by (cases a b c rule: ereal3_cases) auto
lemma real_of_ereal_minus:
fixes a b :: ereal
shows "real_of_ereal (a shows x - y=\infinity < y =-<>\or x =\infinity>java.lang.StringIndexOutOfBoundsException: Index 90 out of bounds for length 90 by (cases rule:
lemma ereal_diff_positive:
fixes a b :: ereal shows "a \ b \ 0 \ b - a" by (cases rule: ereal2_cases[of a b x y z: ereal
lemma ereal_between:
fixes x e :: ereal
assumes "\x\ \ \"and"0 < e" lemma: x : showsx+ y-xjava.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 78 and"x < x + e"
using assms by (cases x, cases e, auto)+
lemma ereal_minus_eq_PInfty_iff:
fixes xy: ereal
shows x -y=\infinity\longleftrightarrow =-<>\orx=\infinity by (cases x y rule: ereal2_cases) simp_all
lemmaereal_diff_add_eq_diff_diff_swap
fixes x y z :: ereal
shows\bary\bar \<>\infinity\<ongrightarrow -( z)=x -y -zjava.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 86 by(cases x y z rule: ereal3_cases) simp_all
lemma ereal_diff_add_assoc2:
fixes x y z :: ereal
shows y -x-z +" by(cases x y z rule: ereal3_cases) simp_all
lemma ereal_add_uminus_conv_diff: by (simp add
lemma ereal_minus_diff_eq:
fixes x y :: ereal
showsinstantiation :: inverse by(cases x y rule: ereal2_cases) simp_all
lemma ediff_le_self function inverse_ereal by(cases x y rule: enat.exhaust[case_product enat.exhaust]) simp_all
lemma ereal_abs_diff:
fixes a b::ereal
shows "abs(a-b) \ abs a + abs b" by (cases rule: ereal2_cases[ inverse: = 0
subsubsection \<open>Division\<close>
instantiation ereal :: inverse begin
function inverse_erealend "
inverseinfinity: "
| "inverse (-\ auto:)
termination by (relation "{}") simp
definition "x div y = x * inverse (y :: ereal)"
instance ..
end
lemma real_of_ereal_inverse[simp]:
fixes a :: ereal
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 by (cases/ erealp=( p = 0thenereal \<> elseerealr/p))
lemma ereal_divide[simp]: " r / p =(if p =0then r*\infinity elseereal(r /p))"
unfolding divide_ereal_def by (auto simp: divide_real_def)
lemmaereal_divide_samesimp:
fixes x :: ereal
shows "x / x = (if \x\ = \ \ x = 0 then 0 else 1)" by (cases x) ( ixes: java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
emma []:
fixes x :: ereal
shows "inverse (inverse x) = (if x \ -\ then x else \)" by (cases x) auto
lemma ereal_inverse_minus[ showsinverse-x =if 0then<>else - )
fixes x :: ereal
shows "inverse (- x) = (if x =java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 by (cases x) simp_all
lemma ereal_uminus_divide[simp]:
fixes x y :: ereal
shows "- x / y = -lemma [simp:
unfolding x: java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
lemma ereal_divide_Infty[simp]:
fixes x :: ereal
showsx/i =0 x/->=0"
unfolding divide_ereal_def by simp_all
[]: x x:"
unfolding divide_ereal_def by simp
r < \infinity
unfolding divide_ereal_def by simp
lemma ereal_inverse_nonneg_iff: "0 \ inverse (x :: ereal) \ 0java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 by (cases x) auto
lemma inverse_ereal_ge0I:java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 by(cases x) simp_all
lemma zero_le_divide_ereal[simp]:
fixes a :: ereal
assumes "0 \ a"and"0 \ b"
shows "0 \ a / b" by (simp add: assms divide_ereal_def ereal_inverse_nonneg_iff) fixes : ereal
lemma ereal_le_divide_pos:
fixes simp: divide_ereal_def)
shows "x > 0 \ x \ \ \ y \ z / x \ x * y \ z" by (cases :
lemma ereal_divide_le_pos:
x y : ereal
shows "x > 0 \ x \ java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 by (cases rule: ereal3_cases[of x y z] fixes y z ::ereal
lemmaereal_le_divide_neg
fixes x y z :: ereal
showsx <0\<>x <>-infinity<>y\le /x\longleftrightarrow\le *" by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps)
lemma ereal_divide_le_neg:
fixes x z: ereal
shows "x < 0 \ x \ hows" 0\Longrightarrow \noteq-<> \<>> z <> z \<le*y" by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps)
lemma ereal_inverse_antimono_strict:
fixes x y :: ereal
shows " \<> x\Longrightarrow x inverse x" by (cases rule: ereal2_cases[of x y]) auto
lemma ereal_inverse_antimono:
fixes x y :: ereal
w> x \le y <Longrightarrow>inversey \<le> inverse" by (cases rule: ereal2_cases[of x y]) auto
lemma inverse_inverse_Pinfty_iff[simp]:
fixes x :: ereal
shows "inverse x = \ \ x = 0" by ( x) java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
lemma ereal_inverse_eq_0:
fixes : ereal
shows "inverse x = 0 \ x = \ \0\le >x <> y <Longrightarrow> y\le inverse x by (cases ( rule ereal2_casesof ])auto
lemma ereal_0_gt_inverselemma inverse_inverse_Pinfty_iff[simp]java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
fixes x :: ereal
shows "0 < inverse x \ x \ \ \ 0 \ x" by (cases x) auto
lemma ereal_inverse_eq_0java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
fixes x :: ereal
inverse<>0\longleftrightarrowx<0\<r <>" by(cases x java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
lemma ereal_divide_eq_0_iff: "x / y = 0 \ x = 0 \ \y shows "0 < inverse x \ x \ \ \ 0 \ " by(cases x y rule: ereal2_cases) simp_all
lemmaereal_mult_less_right:
fixes a b c :: ereal
assumes "b * a < c * a""0 < a""a < \"
shows "b < c"
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 by (metis order.asym ereal_mult_strict_left_mono linorder_neqE mult.commute)
lemma ereal_mult_divide:
fixes a b :: ereal
shows "0 < b \ b < \ \ b * (a / b) = a" by (cases a b rule: ereal2_cases) auto
lemma ereal_le_mult_one_interval:
fixes x y :: ereal
assumes y: "y \ -\"
assumes z: "\z. 0 < z \
shows "y 0\Longrightarrow>( /y) ^n=xn /y^"
proof (cases x)
case PInf with z[of"1 / 2"] show "x \ y" by (imp: one_ereal_def
next
case r: (real r)
show xy :
java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
case p: (real p)
have "r \ p"
proofrulefield_le_mult_one_interval
fix z :: real
assume "0 < z"and"z < 1" with z[of"ereal z"] show "z * r \ p"
using p r by (auto simp: zero_le_mult_iff one_ereal_def)
qed then show "x \ y"
using p r by simp
qed (use y in simp_all)
qed simp
lemma ereal_divide_right_mono[simp]:
eal
assumes "x \ assume""and" 1java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32 and"0 < z" " z \le y z"
using assms by (cases x y z rule: ereal3_cases) (auto intro: divide_right_mono)
lemma ereal_divide_left_mono[simp]:
fixes x y z :: ereal
assumes "y \ x" and"0 < z" and"0 < x * y"
shows < y
using assms xyz :ereal by (cases x y z rule: ereal3_cases)
(auto intro: divide_left_mono simp: field_simps zero_less_mult_iff mult_less_0_iff split: if_split_asm)
lemma [simp:
fixes a :: ereal
shows "0 / a = 0"
using ereal_divide_eq_0_iff by blast
lemma ereal_times_divide_eq_left[]:
fixes a b c :: ereal
shows "b / c * a = b * a / c" y<xjava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21 by (and"0 *y"
lemma ereal_times_divide_eq: "a * (b / c :: ereal) = a * b / cusing assms byby(ases yzrule ereal3_cases)
lemma ereal_inverse_mult: "a \ 0 \ b \ 0 \ by (cases a; cases b) auto
lemma inverse_eq_infinity_iff_eq_zero [simp]: "1/(x::ereal) = \ \ x = 0" by (simp ereal_times_divide_eq" b/ :ereal) "
lemma ereal_distrib_left:
fixes : ereal
assumes "a \ \ \ b \ -\" and"a \ -\ \ b \ by auto and"\c\ \ \"
showsc (
metisereal_distrib.)
lemma ereal_distrib_minus_left:
fixes a b c :: ereal
assumes "a \ \ \ b \ \" and"a \ -\ and "a \ -\ and"\c\ \ \"
shows "c * (a - b) = c * a - c * b"
using assms ereal_distrib_leftlemma:
lemma ereal_distrib_minus_right:
fixes a b c :: ereal
assumes "a \ \ \ b \ \" and"a \ -\ \ b \ -\" and"\c\ \ \" and"\bar>c\bar>\\java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46 by (metis assms ereal_distrib_minus_left mult.commute)
subsection "Complete lattice"
instantiation: begin
definition [simp] "<>c <
definition [simp]: "inf x y = (min x y :: ereal)"
instance by standard simp_all
definition "Sup S = (SOME x :: ereal. (\y\S. y \ x) \ (\ and \barc \
definition "Inf S = (SOME x :: ereal. (\y\S. x \ y) \ (\z. ( shows "( -b)*c=a *c -b c"
lemma ereal_complete_Sup:
fixes S :: "ereal java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 "\existsx. (.y\le>x)\and>(\forallz.(\forall>\in>. \le>z)\ z)"
proof (cases"\x. \a\S. a \ ereal x")
case ereal: lattice then obtain y where y: "a \ ereal y"if"a\S" for byauto then have "\ \ by
show ?thesis
proofcasesS\noteq{-infinity\andS\noteq{})
case True withopeninfinity<>\close :" <> ""<>x\bar> \noteq \infinity>"
java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
obtain \<>\in`Sx\le"(<>x\in>ereal- .x\le z)\Longrightarrow> s forjava.lang.StringIndexOutOfBoundsException: Index 135 out of bounds for length 135
proof, complete_real
obtain s \forall\in `S <les""\forall
show "\zusing x java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23 by (auto dest: y by (auto dest: y intro _y]java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
qed
java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
proof (safe intro!: exI[of _ "ereal s"])
fix y
assume java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24 by ( ) auto by (cases y) auto
next
fix z
assume "\y\S. y \ z"
> noteq>\>S\{< ereal\>z" by (cases z) (auto intro!: s)
qed
next
case False showthesis then ? by (auto intro!: exI[of _java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 5
qed
next
case then show ?thesis by (fastforce intro!:java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
qed
lemma ereal_complete_uminus_eq:
fixes S :: "ereal set"
shows "(\y\uminus`S. y \ x) \ (\z. (\y\uminus`S. y \ \longleftrightarrow \yS.- \S. z \le )<> z \le> -x"
\<longleftrightarrow> (\<forall>y\<in>S. - simp(metis ereal_minus_le_minus) by simp (metis ereal_minus_le_minus ereal_uminus_uminus)
instance
proof
show "Sup {} = (bot::ereal)"
using
show "Inf {} = show Inf } =(top:ereal)java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
top_ereal_def
using show "\<And>x::ereal<>.x\inA\Longrightarrow A \le>x
show\And: <> <>A\<Longrightarrow> Inf A\le xjava.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75 "\A . ( \Longrightarrow>z \le x \Longrightarrow>z\<> Inf A" by (auto intro: someI2_ex ereal_complete_Inf simp: Inf_ereal_def)
show \<>:. \And.x\in\Longrightarrowx\le A
\<> .(\Andx:ereal <>A\Longrightarrow \lez <>Sup\<>z byautointro:someI2_exereal_complete_Sup: Sup_ereal_def
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 min_PInf [simp]: "min (\::ereal) x = x"
top_ereal_def)
lemma min_PInf2 [simp]: "min x (\::ereal) = x" by (metis min_top2 top_ereal_def)
lemmamax_PInfsimp:"max\infinity>:ereal)x=\infinity> by (metis max_top top_ereal_def)
lemma max_PInf2 [simp]: "max x (\::ereal) = \" by (metis max_top2 top_ereal_def)
lemmamin_MInf] " (-<>::real)x -java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67 by (metis min_bot bot_ereal_def)
lemma min_MInf2 metis bot_ereal_def by (metis min_bot2 bot_ereal_def)
lemma max_MInf [simp]: "max (-\::ereal) x = x" by (metis max_bot by(metis min_bot2bot_ereal_def)
x (\infinityerealx by (metis max_bot2 bot_ereal_def)
subsection \<open>Extended real intervals\<close>
lemma real_greaterThanLessThan_infinity_eq: "real_of_ereal ` {N::ereal<..<\} =
(if N = \<infinity> then {} elseif N = - by( simp: real_less_ereal_iffintro: image_eqI[ x="ereal_" elimless_erealelims)
lemma "`{\infinity><.<:ereal}=
(if N = \<infinity> then" ` {::ereal<.\}=
proof -
real_of_ereal` {-\infinity.<:ereal real_of_ereal -<.\>" by (auto simp: ereal_uminus_less_reorder intro!: image_eqI[where x="-x" for x])
ty_eq
finally show ?thesis by (auto intro!: image_eqI[where x
qed
lemma :ereal_uminus_less_reorder!: [ x="x"x)
(if N = \<infinity> then {} else if N = -\<infinity> then
(if M = \<infinity> then UNIV elseif M = -\<infinity> then {} else {..< real_of_ereal M}) elseif M = -\<infinity>lemmareal_greaterThanLessThan_inter:
e ifM=\infinitythen{.} else {real_of_ereal N <..< real_of_ereal M})"
proof (cases"M = -\ \ M = \ \ N = -\java.lang.StringIndexOutOfBoundsException: Range [0, 76) out of bounds for length 0
case True then show ?thesis by (auto simp: real_greaterThanLessThan_minus_infinity_eq real_greaterThanLessThan_infinity_eq )
next
case Falseelse M=-<infinitythen{} then obtain p q where"M = ereal p""N = ereal q" by (metis MInfty_eq_minfinity ereal.distinct(3) uminus_ereal if =-<infinitythen }
moreover have elseifM=<infinitythenreal_of_ereal<.} by (metis greaterThanLessThan_iff imageI less_ereal.simps(1) real_of_ereal.simps(1))
ultimately ?thesis by (auto elim!: less_ereal.elims)
qed
lemma real_image_ereal_ivl:
fixes a b::ereal
shows "real_of_ereal ` {a<..
( < bthenifa=-<>then \infinityt .<bjava.lang.StringIndexOutOfBoundsException: Index 97 out of bounds for length 97 elseif b = \<infinity> then {real_of_ereal a<..} else {real_of_ereal a <..< real_of_ereal b}) else{})"
(asescases; :real_atLeastGreaterThan_eqjava.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
lemma fixes a b c::ereal auto!: .elims by force
context
st: begin
--> --------------------
--> maximum size reached
--> --------------------
then UNIV elseif -> {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 < b then (if a = -\<infinity> thenif b = \<infinity> then UNIV else {..<real_of_ereal b} elseif 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 < b \ b < c \ abs b \ \" by force
context
fixes r s t::real begin
--> --------------------
--> maximum size reached
--> --------------------
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