Theory HoareEx

(*  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 Müller, Nipkow, Oheimb, Slotosch
  citeMuellerNvOS99.  It demonstrates fixpoint reasoning by showing
  the correctness of the Hoare rule for while-loops.

type_synonym assn = "state  bool"

  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