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subsection "A small step semantics on annotated commands"
theory Collecting1
imports Collecting
begin
text\<open>The idea: the state is propagated through the annotated command as an
annotation \<^term>\<open>{s}\<close>, all other annotations are \<^term>\<open>{}\<close>. It is easy
to show that this semantics approximates the collecting semantics.\<close>
lemma step_preserves_le:
"\ step S cs = cs; S' \ S; cs' \ cs \ \
step S' cs' \<le> 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} (annotate (\p. {}) c))"
lemma steps_approx_fix_step: assumes "step S cs = cs" and "s \ S"
shows "steps s (strip cs) n \ cs"
proof-
let ?bot = "annotate (\p. {}) (strip cs)"
have "?bot \ cs" by(induction cs) auto
from step_preserves_le[OF assms(1)_ this, of "{s}"] \<open>s \<in> S\<close>
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
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