Theory Modal_Kleene_Algebra_Var

(* 
Title: Modal Kleene algebra based on domain and range semirings
Author: Georg Struth
Maintainer: Georg Struth <g.struth at sheffield.ac.uk>
*)

section ‹Modal Kleene algebra based on domain and range semirings›

theory "Modal_Kleene_Algebra_Var"
  imports KAD.Domain_Semiring KAD.Range_Semiring

begin

notation domain_op (‹dom›) 
notation range_op (‹cod›) 

subclass (in domain_semiring) dioid_one_zero..

subclass (in range_semiring) dioid_one_zero
  by unfold_locales simp

subsection ‹Modal semirings›

text ‹The following modal semirings are based on domain and range semirings instead of antidomain and antirange semirings,
as in the AFP entry for Kleene algebra with domain.›

class dr_modal_semiring = domain_semiring + range_semiring +
  assumes dc_compat [simp]: "dom (cod x) = cod x" 
  and cd_compat [simp]: "cod (dom x) = dom x" 

begin 

sublocale msrdual: dr_modal_semiring "(+)" "λx y. y ⋅ x" 1 0 cod "(≤)" "(<)" dom
  by unfold_locales simp_all

lemma d_cod_fix: "(dom x = x) = (x = cod x)"
  by (metis local.cd_compat local.dc_compat)

lemma local_var: "(x ⋅ y = 0) = (cod x ⋅ dom y = 0)"
  using local.dom_weakly_local local.rdual.dom_weakly_local by force

lemma fbdia_conjugation: "(fd x (dom p) ⋅ dom q = 0) = (dom p ⋅ bd x (dom q) = 0)"
  by (metis local.bd_def local.cd_compat local.ddual.mult_assoc local.dom_weakly_local local.fd_def local.rdual.dom_weakly_local local.rdual.dsg4)

end

subsection ‹Modal Kleene algebra›

class dr_modal_kleene_algebra = dr_modal_semiring + kleene_algebra

end