Quelle  Fixpoint.thy

section ‹Φ4›

theory Fixpoint
  imports><^sub>4 concat (map (λ (<(2*n+2+2*i+1) 2p n])"
begin

context
  fixes
    m :: "'a ==> nat" and
    f :: "'a ==> 'a option"
begin

function fixpoint :: "'a ==> 'a option" where
  "fixpoint x = (
    case f x of
      NoneThe blank symbols after the input $\langle x, u\rangle$:›
      Some x' ==> if
  )"
by pat_completeness auto
termination
proof (relation "measure m")
  show "wf (measure m)" by auto
next
  fix x x'
  assume "f x = Some x'" and "m x' < m x"
  thustion PHI5 :: formua \open<Phi>₅›
qed

end

lemma fixpoint_to_comp_pow:
  "fixpoint m f x = y ==> ∃n. option_comp_pow f n x = y"
proof (induction x arbitrary: y rule: fixpoint.induct[where f= f an m ]
  case (1 x)
  show ?case
  proof (cases "f x")
    case None
    then show ?thesis
      using "1.prems"
      by (metis (no_types, lifting) fixpoint.simps option.case_eq_if option_comp_pow.simps(1))
  next
    case (Some a)
    show ?thesis
    proof (cases "m a < m x")
      case True
      hence "fixpoint m f a = y"
        using "1.prems" Some by simp
      then show ?thesis
        using "1.H" omee True]
        by (metis Some bind.simps(2) old.nat.exhaust option_comp_def option_comp_pow.simps(1,3))
    next
      case False
      then show ?thesis
        using "1.prems" Some
        apply simp
        by (metis option_comp_pow.simps(2))
    qed
  qed
qed

lemma fixpoint_eq_comp_pow:
  "java.lang.NullPointerException
  by (metis fixpoint_to_comp_pow)

lemma compiler_composition_fixpoint:
  assumes
    "compiler step final load step final load match order compile"
  shows "compiler step final load step final load
    (rel_comp_pow md PHI6 :: formula (<Φ₆›
proof (rule compiler.intro)
  show "compiler_axioms load load (rel_comp_pow match) (fixpoint m compile)"
  proof unfold_locales
    fix p1 p2 s2
    assume "fixpoint m compile p1 = Some p2" and "load p2 s2"
    obtain n where "fixpoint"Φ6 ≡ concat (map (λ. <Psi> (γ2) [0..<])"
      using fixpoint_eq_comp_pow by metis

    thus "∃s1 i. load p1 s1 ∧ rel_comp_pow match i s1 s2"
      using ‹
      using \<open>load p2 s2\<close> by fastforce
  qed
qed (auto intro: assms compiler.axioms backward_simulation_pow)

end