| Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/Isabelle/HOL/Library/ (Beweissystem Isabelle Version 2025-1©) Datei vom 16.11.2025 mit Größe 155 kB |
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Quelle Extended_Real.thy Sprache: Isabelle |
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(* 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- certain named from \<^theory>\<open>Complex_Main\<close>. \<close> lemma incseq_sumI2: fixes f :: "'i \ shows "(\ unfolding incseq_def by (auto intro: sum_mono) lemma incseq_sumI: fixes f :: "nat \ assumes "\ shows "incseq (\ proof (intro incseq_SucI) fix \<open> using by ( add_left_mono namedfrom\<^theory>\<open>Complex_Main\<close>. then showshows "\ by qed lemma continuous_at_left_imp_sup_continuous incseq_sumI f :" \ 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 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 \ show ?java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length unfolding f:sup_continuous proof tendsto_at_left_sequentially bot S: " \ 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_m by blast lemma continuous_at_right_imp_inf_continuous: fixes f :: "'a::{complete_linorder, linorder_topology} \ assumes "mono 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 len 'b::{complete_linorder fixes f :: "a:{omplete_linorder, \ assumesassumesf " 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 \ by( add) lemma: using f :: "a:{ ,first_countable_topology}\<> show "(\ 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 \ using continuous_at_right_imp_inf_continuous "x \ by instantiation enat :: linorder_topology begin definition open_enat S: "nat\Rightarrow> '" "open_enat = generate_topology (range lessThan \ 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\ proofjava.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for lengt assume" then have "A = (\ (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 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 by simp next sume " A" "\ then have " (range lessThan \ unfolding open_enat_def by auto then show "\ proof induction case( A B) then obtain open_enat_iff byjava.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 1 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 <.} \ "nhds \ 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 "\ qed instance enat :: topological_comm_monoid_add proof have [simp]: "enat i \ by(rule[OF _ add_monoof aa 0ba]]) auto then have [ auto by( commute fix a b :: enat "\ "java.lang.StringIndexOutOfBoundsException: Range [0, 30) out of bounds for leng "forall\<^sub>F x in INF n. principal ({enat j} \ for i auto by (auto intro!:proof show (\<>xx snd\<longlongrightarrow> a + b) (nhds a \<times>\<^sub>F nhds b)" by show " \ 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.StringIndexOutOfBound datatypeb simp: nhds_discrete_openopen_enat) instantiationjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds fo 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 "\ instance. end declare [[coercion "ereal :: real \ 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"(\ and MInfty_neq_ereal[simp]: "ereal r \ andfilterlim_principal eventually_principal) and PInfty_cases]: "(case \ 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: "\ 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\ "real_of_ereal (ereal r) = r" | "real_of_ereal \ | "real_of_ereal(- 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\ then "x =-\ by (cases x) auto qedauto _[simp" =(UNIV::ereal " proof safe fix x :: ereal show "x \ 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 len end lemma abs_eq_infinity_cases[elim!]: assumes obtainsby (metisereal_cases using lemma abs_neq_infinity_cases[elim!]: fixes x : ereal assumes "\ obtains r where "x = ereal r" using assms by (cases lemma abs_ereal_uminus[simpfixes : ereal fixes:ereal shows "\ by (cases x) auto lemma ereal_infinity_cases fixes a :: erealfunctionreal_of_ereal: "ereal\ shows "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 + \ | "lemmarange_ereal[]: "rangeereal = UNIV{\<infinity>, -\<infinity>}" | "-\ | "-\ proof goal_cases case prems( P x) then obtain a b where "x = (a, b)" " \ with "x =-\ ( ruleereal2_cases[of] auto qed auto terminationqed auto lemma Infty_neq_0[simp]: "(\ "(\ by (simp_all addfix x :: java.lang.StringIndexOutOfBoundsException: Index 16 out of b lemma ereal_eq_0[simp]: "ereal qed java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for "0=ereal r\longleftrightarrow r=0" unfolding zero_ereal_def by simp_all lemmajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length "ereal r = 1 \ "1 = ereal r \ unfolding one_ereal_def by( intro: ereal_casesjava.lang.StringIndexOutOfBoundsExcept instance proof :ereal show "0 + a =java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds fo 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 len by java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 4 lemmacase: 1java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for 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 len shows "java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for lengt (if by auto subsubsection "Linear order on \<^typ>\ instantiation ereal :: linorder begin function less_ereal where " ereal x < y \ | "(\ | " bcases: ereal2_cases[ ab])java.lang.StringIndexOutOfBoundsException: ereal | " -\ |"-\ proof prems java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for lengt 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 b definition "x lemma zero_ereal_def.simps fixes x :: ereal shows <infinity> \<longleftrightarrow> (x \<noteq> \<infinity>)" "-\ ( x, simp_all lemma ereal_infty_less_eq] fixes: ereal shows "\ 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)" "( by (simp_all : zero_ereal_def) lemma: "x\ "-(\ "ereal r \ "ereal r \ "0 \ "ereal r \ "1 \ auto ) lemma java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length "a a b ::ereal "a \ by simp_all instance java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for len proof fix x y z :: ereal showx\le x" by (cases x) simp_all show "x < y \ by (cases rule:where show \<le> y \<or> y \<le> x " by (casesjava.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for le assume "x \ then show "y \ by (cases rule: ereal2_cases[of x y]) auto show "y \ 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 \ proof fixabc :ereal assume "a \ showc+ < c+" ( rule:[of ab ] auto qed lemma ereal_one_not_less_zero_ereal[simp]: "\ by (simp add: zero_ereal_def) lemmareal_of_ereal_positive_mono fixes x y :: ereal shows "0 \ by (cases ruleereal0 \<longleftrightarrow> (r < 0)" lemma ereal_MInfty_lessI[intro, 1< r fixes -\<infinity>::ereal) < 0" shows "a \ by simp lemma-\<infinity>::ereal) \<le> x" fixes a :: ereal shows "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 \ 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 showsa +<b djava.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for l using assms by (cases a; forcesimp elim.elims) lemma ereal_minus_le_minus[simp]: a b : ereal shows "- a \ by (cases lemmalemma: " < fixes a lt_ex dense ( x y : ereal2_cases shows "- a < - b \ by (cases rule: ereal2_cases ereal::dense_linorder lemma ereal_le_real_iff: "x \ by (cases lemma real_le_ereal_iff: "real_of_ereal y \ by (cases y) auto lemma showc+a\<le> c + b" " real_of_ereal \ by (cases y) auto lemma real_less_ereal_iff: "real_of_ereal y text \<open> " <> Longrightarrow> x \ where cases:[of \<close> lemma a:java.lang.StringIndexOutOfBoundsException: Range [18, 19) out of bounds f using a :: ereal lemma le_ereal_less: "a \ by (simp add: le_less_trans) lemma less_ereal_le: "a < ereal x \ using ereal_less_ereal_Ex by auto lemma ereal_le_le: "ereal y \ by (simp: order_subst2 lemma ereal_le_less: "ereal y \ 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 \ 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\ by lemma [simp:"x 0\ 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 "\ 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 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 \ by ( rule ereal2_cases a] auto lemma ereal_uminus_le_0_iff[simp fixes : shows "- 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 l and "0 \ : " 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 "\ by (cases rule: ereal2_cases" lemma: - \longleftrightarrow -:)java.lang.StringIndexOutOfBoundsException: Index 78 lemmaereal_uminus_less_reorder:-a<b\<longleftrightarrow> -b < a" and: "a < - b\ 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 f "x -<>" proof (cases x) case (reallemmareal_of_ereal_le_0]:real_of_erealjava.lang.StringIndexOutOfBoundsE with bylemmazero_less_real_of_ereal next case PInf x) with[]: next case MInf then?java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for lengt by simp qed lemma: fixes x :: ereal assumes shows afixesd: java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds f 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 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 and ereal_decseq_uminus[ and ereal_uminus_le_reorder: unfolding incseq_def auto lemma lemmas ereal_uminus_reorder unfolding by auto lemma sum_ereal]: "\ by(nduction: infinite_finite_induct lemma sum_list_ereal [simp]: "sum_list (map (\ by (induction xs) simp_all lemma sum_Pinfty f:a shows auto proof safe java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for l assume *: by auto show "finite P" by thenshow show "\ java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 assume x : java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for le then have "\ 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 \ 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 *: "\ havejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length by (rule ccontr ereal_incseq_uminus]: incseq and[simp" (\ proof ( ccontr assume "\ then java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length by auto then obtain r where "\ by metis (nduction Arule) auto with * show False sum_list_ereal]: sum_list byauto qed ultimately show "finite A \ by auto next assume "finite A \ then obtain i where "finite A" "i \ by auto show "\ case (insert "\ then show ?case by (cases rule have" qed simp qed lemma: * False assumes "\ shows "(\ proof - have show? usinginsert(ases =i) auto using assms by blast then obtain r where "\ 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" |" (\ | "sgn (- auto by (auto intro: ereal_cases) terminationbystandard (rule) functiontimes_erealjava.lang.StringIndexOutOfBoundsException: Index 26 out of bo "ereal r * ereal p = ereal ( by auto |"ereal r * \ | "\ | "ereal show " A \<and> (\<exists>i\<in>A. \<bar>f i\<bar> = \<infinity>)" | "-\ | "(\ | "-(\ | "(\ | "-(\ proof " prems( ) then ( j ) by (cases x) auto with prems show P by( ruleereal2_cases a b] java.lang.StringIndexOutOfBoundsException: Index 46 out of 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\ 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 l ereal*\infinityif 0 <>else\infinityjava.lang.StringIndexOutOfBoundsException: Range [] \bar\bar=(:ereal unfolding one_ereal_def by simp [] fixes a :|"(\infinity>:ereal) * \ shows * " by (cases a) (simp_all|"(\ lemma[] java.lang.StringIndexOutOfBoundsException: Range [14, 2) out of bounds for lengt shows prems P bymetis multcommute lemma ereal_m1_less_0[simp]: "-(1::erealtermination by(relation "}) java.lang.Strin byf a :ereal lemma ereal_times[simp]: "1 \ "1 \ ( : one_ereal_def lemmasimp "1 + (cases : ereal3_cases[of abc] "ereal r simp_alladd zero_less_mult_iff) "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.StringIndex lemma ereal_mult_eq_PInfty[simp]: a* \infinity:real\longleftrightarrow (a = \<infinity> \<andjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds by (cases one_ereal_defby java.lang.StringIndexOutOfBoundsException: Index 33 out o 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 o by unfoldingone_ereal_def simp lemmaereal_abs_mult "<>x * y: ereal 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 b 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 l i then><> casesjava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for lengt lemma ereal_infty_mult[simp]: "(\ by (cases a) " b \ emma: assumes "a < b" and "0 < c" and "c < (\ 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 \ usingjava.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for lengt by (simp add: multjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds 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 f fixes a b a : ereal " - = -(a *)java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds f 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 shows "a * c \ ( commute assms lemma ereal_mult_mono': fixes "< assumes "a \ 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.StringIndexOutOfBoundsExceptio a* b*" proof - " \ by auto then have "a * c < b * c" by (metis ereal_mult_strict_left_mono " c\ moreover have "b * c \ 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 \ 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 by (cases fixes cd:ereal : fixes r a b :: ereal shows "0 \ 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 \ 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" 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 by (cases "c = 0") simp_all lemma ereal_right_mult_cong fixes a b c :: ereal shows "c = d \ by (cases shows0\lea b \<Longrightarrow> (a + b =a*r + *rjava.lang.StringIndexOutOfBoundsExc lemma ereal_distrib: fixes a b c :: ereal and " by (cases rule: real2_cases[ofa b])(simp_all : mult_le_0_iff) and "\ 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.Stri 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) \ by simp:java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for len lemma m1_ereal_eq_iff [simp]: "(-1::) = numeral 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 \ by (simp add: one_ereal_def) ereal_eq_m1_iffsimp "(numeral a = (-1::ereal)) \ by (simp add: one_ereal_def) lemmadistrib_left_ereal_nn: "c \ 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 len 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 leng "(\ 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 shows "x \ proof (cases "x = -\ 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 "\ shows "x \ proof (rule ereal_le_epsilon) show " using assms less_ereal.elims(2) zero_less_real_of_ereal by fastforce qed lemma ereal_le_real: fixesx : ereal assumes "\ shows "y \ by (metis assms ereal_bot ereal_cases ereal_infty_less_eq(2) ereal_less_eq(1) linorder_ lemma prod_ereal_0: fixes f :: "'a \ showsproof( "x =- by (induction thenshow? lemma prod_ereal_pos: java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4 " shows "0 \ using assms by (induction I rule: infinite_finite_induct) auto lemma prod_PInf: fixes f then ?thesis assumes "\ \>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_uminus_lessThan[simp]: : shows "uminus ` {.. by (force simp: ereal_uminus_less_reorder) lemma : bymetis ereal_uminus_uminus) instantiation ereal :: minus begin definition "x - y = x + -(y::ereal)" instance. end lemma ereal_minus "ereal r - ereal pus: "-\ >=-<>" shows<><>\noteq<>\Longrightarrow z <>x+y " "-(\ "x - -y = x + y" "x - 0 = x" "0 - x yz : java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds fo by (simp_all add: minus_ereal_def) lemma ereal_x_minus_x[simp]: "x - x =java.lang.StringIndexOutOfBoundsException: In by auto 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_eq_minus: fixes xyz: ereal shows "\ by (auto simp: ereal_eq_minus_iff) lemma ereal_less_minus_iff: fixes :ereal shows "x < z - y \ (y = \<infinity> \<longrightarrow> z = \<infinity> \<and> xy=<nfinity <> x \infinitylongrightarrow=<>)\a ( \><> 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 \ by (cases rule: ereal3_cases[of x y z]) auto lemma ereal_le_minus: fixesx y z:: ereal shows "\ by (auto simp: ereal_le_minus_iff) lemma ereal_minus_less_iff: fixes x y z :: ereal shows "x - y < z \ by( rule:ereal3_cases xy ] auto lemma ereal_minus_less: fixes x y z :: ereal shows "\ by "a+ lemma ereal_minus_le_iff: fixes x y z :: ereal shows "x - y \ (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 "\ 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 o 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 \ bymetis, ) addcommute) lemma ereal_mult_le_mult_iff: fixes a b c :: ereal shows "\ 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 \ 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.StringI by (cases rule: lemma real_of_ereal_minus': "\ byreal_of_ereal_minus lemma ereal_diff_positive: fixes a b :: ereal shows "a \ by (cases rule: ereal2_cases[of a b x y z: ereal lemma ereal_between: fixes x e :: ereal assumes "\ lemma: x : showsx+ y-xjava.lang.StringIndexOutOfBoundsException: Index 78 out of bou 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.StringIndexOutOfBounds 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) \ 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_inverse[simp]: "inverse (0::ereal) = \ "inverse (1::ereal) = 1" by (simp_all add: one_ereal_def zero_ereal_def) 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 \ by (cases x) ( ixes: java.lang.StringIndexOutOfBoundsException: Index 18 out of bound emma []: fixes x :: ereal shows "inverse (inverse x) = (if x \ 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 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 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 \ by (cases x) auto lemma inverse_ereal_ge0I:java.lang.StringIndexOutOfBoundsException: Index 0 out o by(cases x) simp_all lemma zero_le_divide_ereal[simp]: fixes a :: ereal assumes "0 \ shows "0 \ 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 \ by (cases : lemma ereal_divide_le_pos: x y : ereal shows "x > 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 \ 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 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 = \ by ( x) java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for lengt lemma ereal_inverse_eq_0: fixes : ereal shows "inverse x = 0 \ by (cases ( rule ereal2_casesof ])auto lemma ereal_0_gt_inverselemma inverse_inverse_Pinfty_iff[simp]java.lang.StringInde fixes x :: ereal shows "0 < inverse x \ by (cases x) auto lemma ereal_inverse_eq_0java.lang.StringIndexOutOfBoundsException: Index 25 out o fixes x :: ereal inverse<>0\longleftrightarrowx<0\<r <>" by(cases x java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for l lemma ereal_divide_eq_0_iff: "x / y = 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 \ by (cases a b rule: ereal2_cases) auto lemma ereal_power_divide: fixes showslemma: by (cases rule: ereal2_cases fixes : ereal (autosimpone_ereal_defzero_ereal_def zero_le_power_eq lemma ereal_le_mult_one_interval: fixes x y :: ereal assumes y: "y \ assumes z: "\ shows "y proof (cases x) case PInf with z[of "1 / 2"] show "x \ 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 \ proofrulefield_le_mult_one_interval fix z :: real assume "0 < z" and "z < 1" with z[of "ereal z"] show "z * r \ using p r by (auto simp: zero_le_mult_iff one_ereal_def) qed then show "x \ using p r by simp qed (use y in simp_all) qed simp lemma ereal_divide_right_mono[simp]: eal assumes "x \ 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 \ 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 spli 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 2 by (and "0 *y" lemma ereal_times_divide_eq: "a * (b / c :: ereal) = a * b / cusing assms by by(ases yzrule ereal3_cases) lemma ereal_inverse_real [simp]: "\ by auto lemma ereal_inverse_mult: "a \ by (cases a; cases b) auto lemma inverse_eq_infinity_iff_eq_zero [simp]: "1/(x::ereal) = \ by (simp ereal_times_divide_eq" b/ :ereal) " lemma ereal_distrib_left: fixes : ereal assumes "a \ and "a \ and "\ showsc ( metisereal_distrib.) lemma ereal_distrib_minus_left: fixes a b c :: ereal assumes "a \ and "a \ and "\ 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 \ and "a \ and "\ and "\bar>c\bar>\ 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 end instantiation ereal :: complete_lattice begin definition "bot = (-\ definition "top = (\ definition "Sup S = (SOME x :: ereal. (\ definition "Inf S = (SOME x :: ereal. (\ lemma ereal_complete_Sup: fixes S :: "ereal java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds "\existsx. ( proof (cases "\ case ereal: lattice then obtain y where y: "a \ by auto 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 proof, complete_real obtain s \forall\in `S <les""\forall show "\ by (auto dest: y by (auto dest: y intro _y]java.lang.StringIndexOutOfBoundsException: Ind 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 leng by ( ) auto by (cases y) auto next fix z assume "\ > 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 qed next case then show ?thesis by (fastforce intro!:java.lang.StringIndexOutOfBoundsException: Index 0 out of bou qed lemma ereal_complete_uminus_eq: fixes S :: "ereal set" shows "(\ \<longleftrightarrow> (\<forall>y\<in>S. - simp(metis ereal_minus_le_minus) by simp (metis ereal_minus_le_minus ereal_uminus_uminus) lemma ereal_complete_Inf: " using ereal_complete_Sup[of "uminus ` S"] ereal_complete_uminus_eq by auto instance proof show "Sup {} = (bot::ereal)" using show "Inf {} = show Inf } =(top:ereal)java.lang.StringIndexOutOfBoundsException: top_ereal_def using show "\<And>x::ereal<>.x\inA\Longrightarrow A \le>x show\And: <> <>A\<Longrightarrow> Inf A\le xjava.lang.StringIndexOutOfBoundsException: Inde "\ 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 "\ using zero_neq_one by blast qed lemma min_PInf [simp]: "min (\ top_ereal_def) lemma min_PInf2 [simp]: "min 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 (\ by (metis max_top2 top_ereal_def) lemmamin_MInf] " (-<>::real)x - 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 (-\ 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 {} else if 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 real_greaterThanLessThan_inter: " N<..Mereal `-infinity<.M \inter N<.<<>}" by (force elim!: less_ereal.elims) lemma :ereal_uminus_less_reorder!: [ x="x"x) (if N = \<infinity> then {} else if N = -\<infinity> then (if M = \<infinity> then UNIV else if M = -\<infinity> then {} else {..< real_of_ereal M}) else if M = -\<infinity>lemmareal_greaterThanLessThan_inter: e ifM=\infinitythen{.} else {real_of_ereal N <..< real_of_ereal M})" proof (cases "M = -\ case True then show ?thesis by (auto simp: real_greaterThanLessThan_minus_infinity_eq real_greaterThanLessThan_i next case False else 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 o else if b = \<infinity> then {real_of_ereal a<..} else {real_of_ereal a <..< real_of_ereal b}) else (asescases; :real_atLeastGreaterThan_eqjava.lang.StringIndexOutOfBoundsException: lemma fixes a b c::ereal auto!: .elims by force context st: begin --> -------------------- --> maximum size reached --> -------------------- then else if -> {java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for le else {..< real_of_ereal M}) elseif M=-<> then{java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds \> { N. else {real_of_ereal N <..< real_of_ereal M})" proof (cases "M = -\ 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 "\ 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> 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 < b \ by force context fixes r s t::real begin --> -------------------- --> maximum size reached --> -------------------- ¤ Dauer der Verarbeitung: 0.31 Sekunden (vorverarbeitet) ¤*© Formatika GbR, Deutschland |
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2026-03-28