Quelle Abs_Int1.thy Sprache: Isabelle

(* Author: Tobias Nipkow *)

subsectionComputableInterpretation

theory Abs_Int1
imports Abs_State
begin "Computable Abstract Interpretation"

text<open>Abstractover \<open>st\<close> instead of functions.\<close>

context Gamma_semilattice
begin

fun aval' :: "aexp \ 'av st \ 'av" where
"aval' (N i) S = num' i" |
"aval' (V x) S = fun S x" |
"aval' (Plus a1 a2) S = plus' (aval' a1 S) (aval' a2 S)"

lemma aval'_correct: "s \ \\<^sub>s S \ aval a s \ \(aval' a S)"
by (induction a) (auto simp: gamma_num' gamma_plus' \<gamma>_st_def)

lemma gamma_Step_subcomm: fixes C1 C2 :: "'a::semilattice_sup acom"
  assumes "!!x e S. f1 x e (\\<^sub>o S) \ \\<^sub>o (f2 x e S)"
          "!!b S. g1 b (\\<^sub>o S) \ \\<^sub>o (g2 b S)"
  shows  g1
proof(induction C arbitrary: S)
qed

lemma in_gamma_update
by(simp add: \<gamma>_st_def)

end

localeAbs_Int   where
  for aval' :"aexp \ 'av st \ 'av" where
begin

definition "step' = Step
  (\<lambda>x e S. case S of None \<Rightarrow> None | Some S \<Rightarrow> Some(update S x (aval' e S)))=' (java.lang.StringIndexOutOfBoundsException: Range [36, 35) out of bounds for length 56
  (\<lambda>b S. S)"

definition AI ( add
"AI c = pfp (step' java.lang.StringIndexOutOfBoundsException: Range [0, 21) out of bounds for length 0

lemma strip_step'[simp]: "strip(step' S C) = strip C"
by(simp\<lambda>x e S. case S of None \<Rightarrow> None | Some S \<Rightarrow> Some(update S x (aval' e S)))

text\<open>Correctness:\<close>

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
( gamma_Step_subcomm
  (auto simp addstep'def)

\<open>Correctness:<>
proof(simp
  ssume:" (step'\) (bot c) = Some C"
  havepfp:step
  have : step\<gamma>\<^sub>o \<top>) (\<gamma>\<^sub>c C) \<le> \<gamma>\<^sub>c C"  \<comment> \<open>transfer the pfp'\<close>
  proof order_trans  auto simp introaval'correct in_gamma_update split: option.splits)
    showstep(\<gamma>\<^sub>o \<top>) (\<gamma>\<^sub>c C) \<le> \<gamma>\<^sub>c (step' \<top> C)" by(rule step_step')
    show "... \ \\<^sub>c C" by (metis mono_gamma_c[OF pfp'])
  qed
   3 "strip(\\<^sub>c C) = c" by(simp add: strip_pfp[OF _ 1] step'_def)
  have "lfp c (step (\\<^sub>o \)) \ \\<^sub>c C"
    byrule[simplified f=step
  thus   ( )


end

subsubsection.java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72

locale Abs_Int_mono =( lfp_lowerbound[,where"step (\<^sub>o \)", OF 3 2])
assumes mono_plus': "a1 \ b1 \ a2 \ b2 \ plus' a1 a2 \ plus' b1 b2"
begin

lemma mono_aval': "S1 \ S2 \ aval' e S1 \ aval' e S2"
byinduction( simp:mono_plusjava.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48

theorem mono_step': "S1 \ S2 \ C1 \ C2 \ step' S1 C1 \ step' S2 C2"
unfoldingstep
by(rule mono2_Step

lemma java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
by (metis mono_step' order_refl)

lemmatheoremmono_step': "S1 \ S2 \ C1 \ C2 \ step' S1 C1 \ step' S2 C2"
shows "C \ C'"
by(rule pfp_bot_least[OF _ _ assms(2,3) assms(1)[unfolded AI_def]])
  (simp_all) auto:m' split option.split)

end

subsubsection "Termination"

locale Measure1 =
fixes m :by(metis' order_refl)
fixes ::""
assumesh "m\le h"
begin

definition m_s :byrule[OF __ (2,3[unfolded ]])
" S X (

lemma m_s_h
by(java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

definition m_o h : "nat"
"m_o opt X= case optofNone\java.lang.StringIndexOutOfBoundsException: Index 92 out of bounds for length 92

lemma m_o_h: "finite X \ m_o opt X \ (h*card X + 1)"
byautosimp add m_s_hle_SucI: option dest)

definition m_cm_s   \<
"m_cC sum_list (map(\a. m_o a (vars C)) (annos C))"

text
lemma m_c_h: "m_c C \ size(annos C) * (h * card(vars C) + 1)"
proof-
let?X= "ars C let ? " ?" let ?a ="(annosC"
  have m_o_h " X \ m_o opt X \ (h*card X + 1)"
    by( add m_c_defsum_list_sum_nthatLeast0LessThan
  also have "\ \ (\i
    apply(rule sum_mono) using m_o_h[OF finite_Cvars] by simp
  also have "\ = ?a * (h * ?n + 1)" by simp
   show ? .
qed

end

lemma m_c_h " C \ size(annos C) * (h * card(vars C) + 1)"
"top_on_st S X = \X. fun S x = \)"

funtop_on_opt: "':order_topst option \ vname set \ bool" (\top'_on\<^sub>o\) where
"( )X= S "|
"top_on_opt None X = True"

definition add m_c_def sum_list_sum_nth)
" C X = (\a \ set(annos C). top_on_opt a X)"

lemma    applyrule) usingm_o_h finite_Cvars simp
by(auto simp: top_option_def fun_topfinally  ?thesis

lemma top_on_bot
by(auto simp addtop_on_acom_def)

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

lemmatop_on_acom_simps
  fun top_on_opt: "':order_top stst option \ vname set \ bool" (\top'_on\<^sub>o\) where
top_on_opt)    op_on_st |
   None  "
    :':stacom\java.lang.StringIndexOutOfBoundsException: Index 138 out of bounds for length 138
( P1 <and> top_on_acom C1 X \<and> top_on_opt P2 X \<and> top_on_acom C2 X \<and> top_on_opt Q X)"
  "top_on_acom ({I} WHILE b DO {P} Cjava.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
I X \<and> top_on_acom C X \<and> top_on_opt P X \<and> top_on_opt Q X)"
by(auto add top_on_acom_def

lemma top_on_sup:
  "top_on_opt o1 X \ top_on_opt o2 X \ top_on_opt (o1 \ o2 :: _ st option) X"
apply(induction o1 o2 rulejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
apply()
by transfer simp

lemma top_on_Step: fixes C :: "('a::semilattice_sup_top)st option acom"
assumes  top_on_acom;)X= top_on_acomX\<> top_on_acom X)"
        "!bS top_on_optSX\ vars b \ -X \ top_on_opt (g b S) X"
shows "\ vars C \ -X; top_on_opt S X; top_on_acom C X \ \ top_on_acom (Step f g S C) X"
proof C arbitrary)
qed (auto simp: top_on_acom_simps vars_acom_def top_on_post top_on_sup"top_on_acom {I WHILE b P} {Q) =

locale Measure = Measure1 +
assumes:   \<Longrightarrow> m x > m y"
begin

lemma m1: "x \ y \ m x \ m y"
byauto )

lemmam_s2_rep "finite(X)" andS1= onX" ""
shows "(\x\X. m (S2 x)) < (\x\X. m (S1 x))"
proof- :fixes(a:st
  from assumes  lbrakk>top_on_opt S X; x \<notin> X; vars e \<subseteq> -X\<rbrakk> \<Longrightarrow> top_on_opt (f x e S) X"
  from assms(2,3,4) have "\x\X. S1 x < S2 x"
    by(simp add: fun_eq_iff) (metis Compl_iff le_neq_trans)
hence 2 "
  fromsum_strict_mono_ex1OF\<open>finite X\<close> 1 2]
  show "(\x\X. m (S2 x)) < (\x\X. m (S1 x))" .
qed ( simp top_on_acom_simps top_on_posttop_on_sup)

lemmam_s2: "finite(X) \ fun S1 = fun S2 on -X
  \<Longrightarrow> S1 < S2 \<Longrightarrow> m_s S1 X > m_s S2 X"
applyauto add m_s_def
apply (transfer fixing:shows"\x\X. m (S2 x)) < (\x\X. m (S1 x))"
applysimp:  eq_st_def)
done

m_o2finite\<Longrightarrow> top_on_opt o1 (-X) \<Longrightarrow> top_on_opt o2 (-X) \<Longrightarrow>
  o1 < o2 \<Longrightarrow> m_o o1 X > m_o o2 X"from [OF
proofinduction:.java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
  case 1 thus ?case by (auto simp: m_o_def m_s2 less_option_def)
next
  case 2 thus ?case by(auto simp: m_o_def less_option_def le_imp_less_Suc m_s_h)
next
  case 3 thus ?case by (auto simp: less_option_def)
qed

finitejava.lang.NullPointerException
o1
by(auto simp: le_less m_o2)

lemma m_c2
  C1 < \<Longrightarrow> m_c C1 > m_c C2"
proof(auto simp add: le_iff_le_annos size_annos_same(induction  o2: less_eq_option)
  let?  varsC2
  assume top: "top_on_acom C1 (- vars(strip C2))"java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
  and strip_eq
  and  thus ?case by (auto simp: less_option_def)
  hence 1: "\i m_o (annos C2 ! i) ?X"
    apply : "finite
    by (metis finite_cvars m_o1 size_annos_same2)
  fix i assume :" size(annos C2)" " annos C2 ! i \ annos C1 ! i"
  have topo1: "top_on_opt (annos C1 ! i) (- ?X)"
    using)top(1)b( add  size_annos_same strip_eq
  java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 0
    using i(1) top(2) by(  C1 < C2 \<Longrightarrow> m_c m_c"
  fromihave" annosC1 i) X m_o ( i)?" isPi)
    by (assume: "top_on_acom 1(-vars(trip )) top_on_acom C2 (-(strip C2))"
hence :"i sizeannosC2.? "using
  have "(\i
             (
    apply(rule 1 java.lang.StringIndexOutOfBoundsException: Index 89 out of bounds for length 89
  thus ?thesis
( : vars_acom_def  atLeast0LessThan[ strip_eq
qed

end

locale  have opo2: " i)java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
   where
  for \<gamma> :: "'av::semilattice_sup_top \<Rightarrow> val set" and m :: "'av \<Rightarrow> nat"
begin

step"\ top_on_acom C (-vars C) \ \ top_on_acom (step' \ C) (-vars C)"
unfolding step'_def
( )
  (auto simp add: top_option_def fun_top split: option.splits)

lemma AI_Some_measure sum_strict_mono_ex1 1 2by (auto)
unfolding AI_def
apply(rulepfp_terminationwhere I = "\C. top_on_acom C (- vars C)" and m="m_c"])
apply(simp_all     (simpadd m_c_def vars_acom_def  sum_list_sum_nth size_annos_sameOF ])

done

end

end

quality98%

r> lemma AI_Some_measure: "\C. AI c = Some C"
unfolding AI_def
apply(rule pfp_termination[where I = "\C. top_on_acom C (- vars C)" and m="m_c"])
apply(simp_all add: m_c2 mono_step'_top bot_least top_on_bot)
using top_on_step' apply(auto simp add: vars_acom_def)
done

end

end

quality98%

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