Theory HoareEx

theory HoareEx
imports Denotational
(*  Title:      HOL/HOLCF/IMP/HoareEx.thy
    Author:     Tobias Nipkow, TUM
    Copyright   1997 TUM
*)

section "Correctness of Hoare by Fixpoint Reasoning"

theory HoareEx imports Denotational begin

text ‹
  An example from the HOLCF paper by Mueller, Nipkow, Oheimb, Slotosch
  @{cite MuellerNvOS99}.  It demonstrates fixpoint reasoning by showing
  the correctness of the Hoare rule for while-loops.
›

type_synonym assn = "state ⇒ bool"

definition
  hoare_valid :: "[assn, com, assn] ⇒ bool"  ("|= {(1_)}/ (_)/ {(1_)}" 50) where
  "|= {P} c {Q} = (∀s t. P s ∧ D c⋅(Discr s) = Def t ⟶ Q t)"

lemma WHILE_rule_sound:
    "|= {A} c {A} ⟹ |= {A} WHILE b DO c {λs. A s ∧ ¬ bval b s}"
  apply (unfold hoare_valid_def)
  apply (simp (no_asm))
  apply (rule fix_ind)
    apply (simp (no_asm))  "simplifier with enhanced ‹adm›-tactic"
   apply (simp (no_asm))
  apply (simp (no_asm))
  apply blast
  done

end