theory Collecting1
imports Collecting
begin
subsection "A small step semantics on annotated commands"
text{* The idea: the state is propagated through the annotated command as an
annotation @{term "{s}"}, all other annotations are @{term "{}"}. It is easy
to show that this semantics approximates the collecting semantics. *}
lemma step_preserves_le:
"[| step S cs = cs; S' ⊆ S; cs' ≤ cs |] ==>
step S' cs' ≤ cs"
by (metis mono2_step)
lemma steps_empty_preserves_le: assumes "step S cs = cs"
shows "cs' ≤ cs ==> (step {} ^^ n) cs' ≤ cs"
proof(induction n arbitrary: cs')
case 0 thus ?case by simp
next
case (Suc n) thus ?case
using Suc.IH[OF step_preserves_le[OF assms empty_subsetI Suc.prems]]
by(simp add:funpow_swap1)
qed
definition steps :: "state => com => nat => state set acom" where
"steps s c n = ((step {})^^n) (step {s} (anno {} c))"
lemma steps_approx_fix_step: assumes "step S cs = cs" and "s:S"
shows "steps s (strip cs) n ≤ cs"
proof-
let ?bot = "anno {} (strip cs)"
have "?bot ≤ cs" by(induction cs) auto
from step_preserves_le[OF assms(1)_ this, of "{s}"] `s:S`
have 1: "step {s} ?bot ≤ cs" by simp
from steps_empty_preserves_le[OF assms(1) 1]
show ?thesis by(simp add: steps_def)
qed
theorem steps_approx_CS: "steps s c n ≤ CS c"
by (metis CS_unfold UNIV_I steps_approx_fix_step strip_CS)
end