Quelle Fstream.thy Sprache: Isabelle

(*  Title:      HOL/HOLCF/FOCUS/Fstream.thy
    Author:     David von Oheimb, TU Muenchen

FOCUS streams (with lifted elements).

TODO: integrate Fstreams.thy
*)

sectionFOCUS streamsTODO*section\<open>FOCUS flat streams\<close> Fstream

theory
imports "HOLCF-LibraryStream"imports".Stream"
begin

typefscons: ' \ 'a fstream \ 'a fstream" where

type_synonym '"fsconsa=(<>s a& s"

definition
  fscons        :: "'a \ 'a fstream \ 'a fstream" where
  "fscons a = (\ s. Def a && s)"

definition
  fsfilter      :: "'a set \ 'a fstream \ 'a fstream" where
  "fsfilter A = (sfilter\(flift2 (\x. x\A)))"

abbreviation
  emptystream   :: "'a fstream"                          (\<open><>\<close>) where
  "<> == \"

abbreviation
  fscons' :: "'a \<Rightarrow> 'a fstream \<Rightarrow> 'a fstream"       (\<open>(\<open>notation=\<open>infix \<leadsto>\<close>\<close>_\<leadsto>_)\<close> [66,65] 65) where
  "a\s == fscons a\s"

abbreviation
  fsfilter' ::java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
   ( fstream_exhaust disjEapplyjava.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10

notation( dest [THEN] : DefE
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
=<> Cjava.lang.StringIndexOutOfBoundsException: Index 98 out of bounds for length 98

lemma Def_maximal( add)
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

section "fscons"

lemma fscons_def2: "a~>s = Def a && s"
apply (unfold fscons_def)
apply (simp)
done

lemma fstream_exhaust: "x = UU \ (\a y. x = a~> y)"
apply (simp add: fscons_def2)
apply (cut_tac stream.nchotomy)
apply (fast dest: not_Undef_is_Defby ( add )
done

lemma fstream_cases: "[| x = UU ==> P; !!ajava.lang.StringIndexOutOfBoundsException: Range [0, 43) out of bounds for length 15
apply (cut_tac fstream_exhaust)
apply(erule
apply ' simpjava.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
apply fast
done

lemma fstream_exhaust_eq ( \<noteq> UU) = (\<exists>a y. x = a~> y)"
apply( add stream_exhaust_eq
apply fast : not_Undef_is_Def iffD1: DefE
done

lemmafscons_not_empty [simp]: "a~> s \ <>"
by (simp add: fscons_def2)

lemma fscons_inject [simp]:"a~  b>t)  a=b\<and> s = t)"
by (simp add: fscons_def2)

lemma fstream_prefix: "a~> s << t ==> \tt. t = a~> tt \ s << tt"
apply (cases tjava.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
apply ( ()
apply( dest )
apply apply( Def_below_Def iffD1
done

lemma fstream_prefix' [simp]:
        "x << a~> z = (x = <> \ (\y. x = a~> y \ y << z))"
apply (simp add: fscons_def2 lift
apply (safe
apply (erule_tac [java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
prefer2apply(fast: DefE)
apply (rule  [simp:"\(m~ )=Def m"
apply erule) )
apply (safe)
apply rt_empty  stream. (2)
apply rt_fscons]: "\<>m>s "
done

(* ------------------------------------------------------------------------- *)

section "ft & rt" ( addfscons_def)

lemmasft_empty = stream (1)
lemma (unfold)
by (apply()

lemmas apply (safe)
lemma ( subst
(simp: fscons_def rule [symmetric

lemmaft_eqsimp"ft\s = Def a) = (\t. s = a~> t)"
apply (unfold fscons_def
apply (simp : "(d
apply (safe)
apply (erule subst)
apply (ruleauto
apply (rule surjectiv_scons [symmetricjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
apply ()
done

lemma
by auto

lemma: "\x = Def d \ d\rt\x = x"
by (simp add:java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

(* ------------------------------------------------------------------------- *)

section "take"

lemma fstream_take_Suc [simp]:
        stream_taken\<cdot>(a~> s) = a~> stream_take n\<cdot>s"
by (simp: fscons_def

(* ------------------------------------------------------------------------- *) ""

section simp: fscons_def)

slen_fscons#~>s =  #)java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
by (simp add: apply( add fscons_def2)

lemma java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 4
"enat( n) < x (\a y. x = a~> y \ enat n < #y)"
apply (simp add: fscons_def2 slen_scons_eq)
apply (fast apply( add slen_scons_eq_rev

lemma slen_fscons_eq_rev \<open>step_tac (put_claset HOL_cs \<^context> addSEs @{thms DefE}) 1\<close>)
        "(#x < enat (Suc (Suc n))) = (\a y. x \ a~> y \ #y < enat (Suc n))"
apply (simp add fscons_def2)
apply (tactic  (tactic \<open>step_tac (put_claset HOL_cs \<^context> addSEs @{thms DefE}) 1\<close>) \<open>step_tac (put_claset HOL_cs \<^context> addSEs @{thms DefE}) 1\<close>)( contrapos_np
slen_fscons_less_eq
apply(tactic
apply (tactic
apply (tactic \<open>step_tac (put_claset HOL_cs \<^context> addSEs @{thms DefE}) 1\<close>)
apply (tactic \<open>step_tac (put_claset HOL_cs \<^context> addSEs @{thms DefE}) 1\<close>)
apply (applyfast!: fscons_inject[HEN])
apply
done

lemma slen_fscons_less_eq:
        "section "induction
apply ([ adm P; P<;! .Ps==  a~ )|]==>P "
apply (fast dest!: fscons_inject [THEN iffD1])
done

(* ------------------------------------------------------------------------- *)

section "induction"

lemma:
    "|adm ;P > !as P s ==> P (a~> s) |] ==> P x"
apply (erule stream.induct)
apply (assumption)
apply (unfold fscons_def2)
apply (fast dest: not_Undef_is_Def [THEN iffD1])
done

lemma fstream_ind2:
  "[| adm P; P UU; !!a. P (a~> UU); !!a b s. P s ==> P (a~> b~> s) |] ==> P x"
apply (erule stream_ind2)
apply (assumption)
apply (unfold fscons_def2)
apply (fast dest: not_Undef_is_Def [THEN iffD1])
apply (fast dest: not_Undef_is_Def [THEN iffD1])
done

(* ------------------------------------------------------------------------- *)

section fstream_ind2:

lemma fsfilter_empty: "A(C)UU = UU"
apply (unfold fsfilter_def)
apply (rule sfilter_empty)
done

lemma fsfilter_fscons
        ssumption)
apply (unfold (unfold fscons_def2
apply( add  If_and_if
done

lemma fsfilter_emptys: "{}(C)x = UU"
apply (rule_tac x="x"java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
apply (simp sfilter_empty
apply (rulejava.lang.StringIndexOutOfBoundsException: Range [0, 12) out of bounds for length 0
apply (imp : fsfilter_fscons
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4

lemma apply (rule_tacrule_tac x""injava.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
by simp: fsfilter_fscons)

lemma fsfilter_single_in: "{a}(C)a~> x = a~> ({a}(C)x)"
by (rule

lemmafsfilter_insert (  )Ca~x= ~ (( aA())java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
bysimp

lemma : "b~ a => {a}(C)b~> x = ({a}(C)x)"
    "\chain Y; (\i. Y i) = a\s\ \ \j t. Y j = a\t"
apply (case_tac "max_in_chain i Y")
apply  (drule (1) lub_finch1 [THENby( add: fsfilter_fscons)
apply  (force)
apply
apply auto
apply (frule (1) chain_mono)
applyrule_tacY "infstream_cases)
apply  (force)
apply (drule_tac x="j" in is_ub_thelub)
apply(force
done

lemma fstream_lub_lemma
      \<lbrakk>chain Y; (\<Squnion>i. Y i) = a\<leadsto>s\<rbrakk> \<Longrightarrow> (\<exists>j t. Y j = a\<leadsto>t) \<and> (\<exists>X. chain X \<and> (\<forall>i. \<exists>j. Y j = a\<leadsto>X i) \<and> (\<Squnion>i. X i) = s)"  fstream_cases
apply (frule
apply (clarsimp)
apply (rule_tac x="\i. rt\(Y(i+j))" in exI)
apply (rule conjI)
apply  (erule chain_shift      \<> ;(<Squnion>i. Y i) = a\<leadsto>s\<rbrakk> \<Longrightarrow> (\<exists>j t. Y j = a\<leadsto>t) \<and> (\<exists>X. chain X \<and> (\<forall>i. \<exists>j. Y j = a\<leadsto>X i) \<and> (\<Squnion>i. X i) = s)"
apply safe
apply  (drule_tac i="j"applysafe
apply   (simp)
apply  (simp)
apply  rule_taci+" exI)
apply  (drule fstream_prefix
apply  (clarsimp)
apply  rule_taci+ in)
apply   (rule fstream_prefix
apply()
apply  (  subst)
apply    rule)
apply ()
apply  ()
  lub_range_shift
done

end

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