Theory HS_VC_KA_ndfun
subsection ‹ State transformer model ›
text ‹ We show that Kleene algebras have a state transformer model. For this we use the type of
non-deterministic functions of the Transformer\_Semantics.Kleisli\_Quantale theory. Below we
prove some auxiliary lemmas for them and show this instantiation. ›
theory HS_VC_KA_ndfun
imports
Kleene_Algebra.Kleene_Algebra
"Transformer_Semantics.Kleisli_Quantale"
begin
notation Abs_nd_fun (‹_⇧∙› [101] 100)
and Rep_nd_fun (‹_⇩∙› [101] 100)
declare Abs_nd_fun_inverse [simp]
lemma nd_fun_ext: "(⋀x. (f⇩∙) x = (g⇩∙) x) ⟹ f = g"
apply(subgoal_tac "Rep_nd_fun f = Rep_nd_fun g")
using Rep_nd_fun_inject
apply blast
by(rule ext, simp)
lemma nd_fun_eq_iff: "(f = g) = (∀x. (f⇩∙) x = (g⇩∙) x)"
by (auto simp: nd_fun_ext)
instantiation nd_fun :: (type) kleene_algebra
begin
definition "0 = ζ⇧∙"
definition "star_nd_fun f = qstar f" for f::"'a nd_fun"
definition "f + g = ((f⇩∙) ⊔ (g⇩∙))⇧∙"
thm sup_nd_fun_def sup_fun_def
named_theorems nd_fun_ka "kleene algebra properties for nondeterministic functions."
lemma nd_fun_plus_assoc[nd_fun_ka]: "x + y + z = x + (y + z)"
and nd_fun_plus_comm[nd_fun_ka]: "x + y = y + x"
and nd_fun_plus_idem[nd_fun_ka]: "x + x = x" for x::"'a nd_fun"
unfolding plus_nd_fun_def by (simp add: ksup_assoc, simp_all add: ksup_comm)
lemma nd_fun_distr[nd_fun_ka]: "(x + y) ⋅ z = x ⋅ z + y ⋅ z"
and nd_fun_distl[nd_fun_ka]: "x ⋅ (y + z) = x ⋅ y + x ⋅ z" for x::"'a nd_fun"
unfolding plus_nd_fun_def times_nd_fun_def by (simp_all add: kcomp_distr kcomp_distl)
lemma nd_fun_plus_zerol[nd_fun_ka]: "0 + x = x"
and nd_fun_mult_zerol[nd_fun_ka]: "0 ⋅ x = 0"
and nd_fun_mult_zeror[nd_fun_ka]: "x ⋅ 0 = 0" for x::"'a nd_fun"
unfolding plus_nd_fun_def zero_nd_fun_def times_nd_fun_def by auto
lemma nd_fun_leq[nd_fun_ka]: "(x ≤ y) = (x + y = y)"
and nd_fun_less[nd_fun_ka]: "(x < y) = (x + y = y ∧ x ≠ y)"
and nd_fun_leq_add[nd_fun_ka]: "z ⋅ x ≤ z ⋅ (x + y)" for x::"'a nd_fun"
unfolding less_eq_nd_fun_def less_nd_fun_def plus_nd_fun_def times_nd_fun_def sup_fun_def
by (unfold nd_fun_eq_iff le_fun_def, auto simp: kcomp_def)
lemma nd_star_one[nd_fun_ka]: "1 + x ⋅ x⇧⋆ ≤ x⇧⋆"
and nd_star_unfoldl[nd_fun_ka]: "z + x ⋅ y ≤ y ⟹ x⇧⋆ ⋅ z ≤ y"
and nd_star_unfoldr[nd_fun_ka]: "z + y ⋅ x ≤ y ⟹ z ⋅ x⇧⋆ ≤ y" for x::"'a nd_fun"
unfolding plus_nd_fun_def star_nd_fun_def
apply(simp_all add: fun_star_inductl sup_nd_fun.rep_eq fun_star_inductr)
by (metis order_refl sup_nd_fun.rep_eq uwqlka.conway.dagger_unfoldl_eq)
instance
apply intro_classes
using nd_fun_ka by simp_all
end
end