header "Correctness of Hoare by Fixpoint Reasoning"
theory HoareEx imports Denotational begin
text {*
An example from the HOLCF paper by Müller, Nipkow, Oheimb, Slotosch
\cite{MuellerNvOS99}. It demonstrates fixpoint reasoning by showing
the correctness of the Hoare rule for while-loops.
*}
types assn = "state => bool"
definition
hoare_valid :: "[assn, com, assn] => bool" ("|= {(1_)}/ (_)/ {(1_)}" 50) where
"|= {A} c {B} = (∀s t. A s ∧ D c $(Discr s) = Def t --> B t)"
lemma WHILE_rule_sound:
"|= {A} c {A} ==> |= {A} \<WHILE> b \<DO> c {λs. A s ∧ ¬ b s}"
apply (unfold hoare_valid_def)
apply (simp (no_asm))
apply (rule fix_ind)
apply (simp (no_asm)) -- "simplifier with enhanced @{text adm}-tactic"
apply (simp (no_asm))
apply (simp (no_asm))
apply blast
done
end