Theory Two_Kleene_Algebra

(* 
Title: 2-Kleene Algebras
Author: Cameron Calk, Georg Struth
Maintainer: Georg Struth <g.struth at sheffield.ac.uk>
*)

section ‹2-Kleene algebras›

theory "Two_Kleene_Algebra"
  imports Quantales_Converse.Modal_Kleene_Algebra_Var

begin

text ‹Here we develop (globular) 2-semigroups and (globular) 2-Kleene algebras. These should eventually
be extended to n-structures and omega-structures.›

subsection ‹Copies for 0-structures›

class comp0_op =
  fixes comp0 :: "'a ⇒ 'a ⇒ 'a" (infixl "⋅0" 70)

class id0_op =
  fixes id0 :: "'a" ("10")

class star0_op =
  fixes star0 :: "'a ⇒ 'a"

class dom0_op =
  fixes dom0 :: "'a ⇒ 'a"

class cod0_op =
  fixes cod0 :: "'a ⇒ 'a"

class monoid_mult0 = comp0_op + id0_op +
  assumes par_assoc0: "x ⋅0 (y ⋅0 z) = (x ⋅0 y) ⋅0 z"
  and comp0_unl: "10 ⋅0 x = x" 
  and comp0_unr: "x ⋅0 10 = x" 

class dioid0 = monoid_mult0 + join_semilattice_zero +
  assumes distl0: "x ⋅0 (y + z) = x ⋅0 y + x ⋅0 z"
  and distr0: "(x + y) ⋅0 z = x ⋅0 z + y ⋅0 z"
  and annil0: "0 ⋅0 x = 0"
  and annir0: "x ⋅0 0 = 0"

class kleene_algebra0 = dioid0 + star0_op +
  assumes star0_unfoldl: "10 + x ⋅0 star0 x ≤ star0 x"  
  and star0_unfoldr: "10 + star0 x ⋅0 x ≤ star0 x"
  and star0_inductl: "z + x ⋅0 y ≤ y ⟹ star0 x ⋅0 z ≤ y"
  and star0_inductr: "z + y ⋅0 x ≤ y ⟹ z ⋅0 star0 x ≤ y"

class domain_semiring0 = dioid0 + dom0_op +
  assumes d0_absorb: "x ≤ dom0 x ⋅0 x"
  and d0_local: "dom0 (x ⋅0 dom0 y) = dom0 (x ⋅0 y)"
  and d0_add: "dom0 (x + y) = dom0 x + dom0 y"
  and d0_subid: "dom0 x ≤ 10"
  and d0_zero: "dom0 0 = 0"

class codomain_semiring0 = dioid0 + cod0_op +
  assumes cod0_absorb: "x ≤ x ⋅0 cod0 x" 
  and cod0_local: "cod0 (cod0 x ⋅0 y) = cod0 (x ⋅0 y)"
  and cod0_add: "cod0 (x + y) = cod0 x + cod0 y"
  and cod0_subid: "cod0 x ≤ 10"
  and cod0_zero: "cod0 0 = 0"

class modal_semiring0 = domain_semiring0 + codomain_semiring0 +
  assumes dc_compat0: "dom0 (cod0 x) = cod0 x" 
  and cd_compat0:  "cod0 (dom0 x) = dom0 x" 

class modal_kleene_algebra0 = modal_semiring0 + kleene_algebra0

sublocale monoid_mult0 ⊆ mm0: monoid_mult "10" "(⋅0)"
  by (unfold_locales, simp_all add: local.par_assoc0 local.comp0_unl local.comp0_unr)

sublocale dioid0 ⊆ d0: dioid_one_zero _ "(⋅0)" "10" _ _ _
  by (unfold_locales, simp_all add: local.distl0 local.distr0 annil0 annir0)

sublocale dioid0 ⊆ dd0: dioid0 _ _ _ _ "λx y. y ⋅0 x" _
  by unfold_locales (simp_all add: local.mm0.mult_assoc local.d0.distrib_left)

sublocale kleene_algebra0 ⊆ k0: kleene_algebra _ "(⋅0)" "10" _ _ _ star0
  apply unfold_locales
  using local.star0_unfoldl apply blast
  by (simp_all add: local.star0_inductl local.star0_inductr local.star0_unfoldl)

sublocale kleene_algebra0 ⊆ dk0: kleene_algebra0 _ _ _ _ "λx y. y ⋅0 x" _ _
  by (unfold_locales, simp_all add: local.star0_inductr local.star0_inductl)

sublocale domain_semiring0 ⊆ dsr0: domain_semiring _ "(⋅0)" "10" _ "dom0" _ _
  apply unfold_locales
      apply (simp add: local.d0_absorb local.join.sup_absorb2)
     apply (simp add: local.d0_local)
    apply (simp add: local.d0_subid local.join.sup_absorb2)
  apply (simp add: local.d0_zero)
  by (simp add: local.d0_add)

sublocale codomain_semiring0 ⊆ csr0: range_semiring  _ "(⋅0)" "10" _ "cod0" _ _
  apply unfold_locales
      apply (simp add: local.cod0_absorb local.join.sup_absorb2)
     apply (simp add: local.cod0_local)
    apply (simp add: local.cod0_subid local.join.sup_absorb2)
   apply (simp add: local.cod0_zero)
  by (simp add: local.cod0_add)

sublocale codomain_semiring0 ⊆ ds0dual: domain_semiring0 _ _ _ _ "λx y. y ⋅0 x" _ cod0
  by unfold_locales simp_all

sublocale modal_semiring0 ⊆ msr0: dr_modal_semiring _ "(⋅0)" "10" _  "dom0" _ _ "cod0"
  by (unfold_locales, simp_all add: local.dc_compat0 local.cd_compat0)

sublocale modal_semiring0 ⊆ msr0dual: modal_semiring0 "dom0" _ _ _ _ "λx y. y ⋅0 x" _ "cod0"
  by unfold_locales simp_all

sublocale modal_kleene_algebra0 ⊆ mka0: dr_modal_kleene_algebra _ "(⋅0)" "10" _ _ _ star0  "dom0" "cod0"..

sublocale modal_kleene_algebra0 ⊆ mka0dual: modal_kleene_algebra0 _ _ _ _ "λx y. y ⋅0 x" _ _ "dom0" "cod0"..


subsection ‹Copies for 1-structures›

class comp1_op =
  fixes comp1 :: "'a ⇒ 'a ⇒ 'a" (infixl "⋅1" 70)

class id1_op =
  fixes id1 :: "'a" ("11")

class star1_op =
  fixes star1 :: "'a ⇒ 'a"

class dom1_op =
  fixes dom1 :: "'a ⇒ 'a"

class cod1_op =
  fixes cod1 :: "'a ⇒ 'a"

class monoid_mult1 = comp1_op + id1_op +
  assumes par_assoc1: "x ⋅1 (y ⋅1 z) = (x ⋅1 y) ⋅1 z"
  and comp1_unl: "11 ⋅1 x = x" 
  and comp1_unr: "x ⋅1 11 = x"

class dioid1 = monoid_mult1 + join_semilattice_zero +
  assumes distl1: "x ⋅1 (y + z) = x ⋅1 y + x ⋅1 z"
  and distr1: "(x + y) ⋅1 z = x ⋅1 z + y ⋅1 z"
  and annil1: "0 ⋅1 x = 0"
  and annir1: "x ⋅1 0 = 0"

class kleene_algebra1 = dioid1 + star1_op +
  assumes star1_unfoldl: "11 + x ⋅1 star1 x ≤ star1 x"  
  and star1_unfoldr: "11 + star1 x ⋅1 x ≤ star1 x"
  and star1_inductl: "z + x ⋅1 y ≤ y ⟹ star1 x ⋅1 z ≤ y"
  and star1_inductr: "z + y ⋅1 x ≤ y ⟹ z ⋅1 star1 x ≤ y"

class domain_semiring1 = dioid1 + dom1_op +
  assumes d1_absorb: "x ≤ dom1 x ⋅1 x"
  and d1_local: " dom1 (x ⋅1 dom1 y) = dom1 (x ⋅1 y)"
  and d1_add: "dom1 (x + y) = dom1 x + dom1 y"
  and d1_subid: "dom1 x ≤ 11"
  and d1_zero: "dom1 0 = 0"

class codomain_semiring1 = dioid1 + cod1_op +
  assumes cod1_absorb: "x ≤ x ⋅1 cod1 x" 
  and cod1_local: "cod1 (cod1 x ⋅1 y) = cod1 (x ⋅1 y)"
  and cod1_add: "cod1 (x + y) = cod1 x + cod1 y"
  and cod1_subid: "cod1 x ≤ 11"
  and cod1_zero: "cod1 0 = 0"

class modal_semiring1 = domain_semiring1 + codomain_semiring1 +
  assumes dc_compat1: "dom1 (cod1 x) = cod1 x" 
  and cd_compat1:  "cod1 (dom1 x) = dom1 x" 

class modal_kleene_algebra1 = modal_semiring1 + kleene_algebra1

sublocale  monoid_mult1 ⊆ mm1: monoid_mult "11" "(⋅1)"
  by (unfold_locales, simp_all add: local.par_assoc1 comp1_unl comp1_unr) 

sublocale  dioid1 ⊆d1: dioid_one_zero  _ "(⋅1)" "11" _ _ _ 
  by (unfold_locales, simp_all add: local.distl1 local.distr1 local.annil1 local.annir1)

sublocale  dioid1 ⊆ dd1: dioid1 _ _ _ _ "λx y. y ⋅1 x"  "11"
  apply unfold_locales
        apply simp_all
   apply (simp add: local.mm1.mult_assoc)
  by (simp add: local.d1.distrib_left)

sublocale kleene_algebra1 ⊆ k1: kleene_algebra _ "(⋅1)" "11" _ _ _ star1
  apply unfold_locales
  using local.star1_unfoldl apply blast
   apply (simp add: local.star1_inductl)
  by (simp add: local.star1_inductr)

sublocale kleene_algebra1 ⊆ dk1: kleene_algebra1 _ _ _ _ "λx y. y ⋅1 x" "11" star1
  by (unfold_locales, simp_all add: local.star1_inductr local.star1_inductl)

sublocale domain_semiring1 ⊆ dsr1: domain_semiring _ "(⋅1)" "11" _ "dom1" _ _ 
  apply unfold_locales
  using local.d1_absorb local.join.sup_absorb2 apply force
     apply (simp add: local.d1_local)
  using local.d1_subid local.join.sup_absorb2 apply force
  using local.d1_zero apply fastforce
  by (simp add: local.d1_add)

sublocale codomain_semiring1 ⊆ csr1: range_semiring _ "(⋅1)" "11" _ cod1 _ _
  apply unfold_locales
      apply (simp add: local.cod1_absorb local.join.sup_absorb2)
     apply (simp add: local.cod1_local)
    apply (simp add: local.cod1_subid local.join.sup_absorb2)
  using local.cod1_zero apply fastforce
  by (simp add: local.cod1_add)

sublocale codomain_semiring1 ⊆ ds1dual: domain_semiring1 _ _ _ _ "λx y. y ⋅1 x" _ cod1
  by (unfold_locales, simp_all add: local.cod1_absorb local.cod1_local local.cod1_add local.cod1_subid)

sublocale modal_semiring1 ⊆ msr1: dr_modal_semiring _ "(⋅1)" "11" _   "dom1" _ _ "cod1"
  apply unfold_locales
   apply (simp add: local.dc_compat1)
  by (simp add: local.cd_compat1)

sublocale modal_semiring1 ⊆ msr1dual: modal_semiring1 "dom1" _ _ _ _ "λx y. y ⋅1 x" _ "cod1"
  by unfold_locales simp_all

sublocale modal_kleene_algebra1 ⊆ mka1: dr_modal_kleene_algebra _ "(⋅1)" "11" _ _ _ star1 "dom1" "cod1"..

sublocale modal_kleene_algebra1 ⊆ mka1dual: modal_kleene_algebra1 _ _ _ _ "λx y. y ⋅1 x" _ _ "dom1" "cod1"..

subsection ‹Globular 2-semirings›

class two_semiring = modal_semiring0 + modal_semiring1 +
  assumes interchange: "(w ⋅1 x) ⋅0 (y ⋅1 z) ≤ (w ⋅0 y) ⋅1 (x ⋅0 z)"
  and d1_hom: "dom1 (x ⋅0 y) ≤ dom1 x ⋅0 dom1 y"
  and c1_hom: "cod1 (x ⋅0 y) ≤ cod1 x ⋅0 cod1 y"
  and d0_hom: "dom0 (x ⋅1 y) ≤ dom0 x ⋅1 dom0 y"
  and c0_hom: "cod0 (x ⋅1 y) ≤ cod0 x ⋅1 cod0 y"
  and d1d0 [simp]: "dom1 (dom0 x) = dom0 x"

class strong_two_semiring = two_semiring + 
  assumes d1_strong_hom [simp]: "dom1 (x ⋅0 y) = dom1 x ⋅0 dom1 y"
  and c1_strong_hom: "cod1 (x ⋅0 y) = cod1 x ⋅0 cod1 y"

sublocale two_semiring ⊆ tgsdual: two_semiring "dom0" _ _ _ _ "λx y. y ⋅0 x" _ "cod0"  "dom1" "λx y. y ⋅1 x" _ "cod1"
  apply unfold_locales
       apply (simp_all add: local.interchange local.d0_hom local.c0_hom local.c1_hom local.d1_hom)
  by (metis local.cd_compat1 local.d1d0 local.dc_compat0)

sublocale strong_two_semiring ⊆ stgsdual: strong_two_semiring "dom0" _ _ _ _ "λx y. y ⋅0 x" _ "cod0"  "dom1" "λx y. y ⋅1 x" _ "cod1"
  apply unfold_locales by (simp_all add: local.c1_strong_hom)

context two_semiring
begin
  
lemma c1d0 [simp]: "cod1 (dom0 x) = dom0 x"
proof-
  have "cod1 (dom0 x) = cod1 (dom1 (dom0 x))"
    by simp
  also have "… = dom1 (dom0 x)"
    using local.cd_compat1 by presburger
  also have "… = dom0 x"
    by simp
  finally show ?thesis.
qed

lemma d1c0 [simp]: "dom1 (cod0 x) = cod0 x"
  by (simp add: msr1.d_cod_fix)

lemma c1c0 [simp]: "cod1 (cod0 x) = cod0 x"
  by simp

lemma "11 ⋅0 11 ≤ 11"
  (*nitpick [expect=genuine]*)
  oops

lemma id1_comp0_var: "11 ≤ 11 ⋅0 11"
proof-
  have "11 = 11 ⋅0 10"
    by simp
  also have "… = (11 ⋅1 11) ⋅0 (10 ⋅1 11)"
    by simp
  also have "… ≤ (11 ⋅0 10) ⋅1 (11 ⋅0 11)"
    using local.interchange by presburger
  also have "… = 11 ⋅1 (11 ⋅0 11)"
    by simp
  also have "… = 11 ⋅0 11"
    by simp
  finally show ?thesis.
qed

lemma "11 ⋅0 11 = 11"
  (*nitpick*)
  oops

lemma id0_le_id1: "10 ≤ 11"
proof-
  have "10 = 10 ⋅0 10"
    by simp
  also have "... = (11 ⋅1 10) ⋅0 (10 ⋅1 11)"
    by simp
  also have "... ≤ (11 ⋅0 10) ⋅1 (10 ⋅0 11)"
    using local.interchange by blast
  also have "... = 11 ⋅1 11"
    by simp
  also have "... = 11"
    by simp
  finally show ?thesis.
qed

lemma id0_comp1_eq [simp]: "10 ⋅1 10 = 10"
proof-
  have "10 ⋅1 10 = dom0 10 ⋅1 dom0 10"
    by simp
  also have "… = dom1 (dom0 10) ⋅1 dom0 10"
    using local.d1d0 by presburger
  also have "… = dom0 10"
    by simp
  finally show ?thesis
    by simp
qed

lemma d1_id0 [simp]: "dom1 10 = 10"
proof-
  have "dom1 10 = dom1 (dom0 10)"
    by simp
  also have "… = dom0 10"
    using local.d1d0 by blast
  also have "… = 10"
    by simp
  finally show ?thesis.
qed

lemma d0_id1 [simp]: "dom0 11 = 10"
proof-
  have a: "dom0 11 ≤ 10"
    by simp
  have "10 ≤ 11"
    by (simp add: local.id0_le_id1)
  hence "10 ≤ dom0 11"
    using local.dsr0.d_iso by fastforce
  thus ?thesis
    by (simp add: local.dual_order.antisym)
qed

lemma c0_id1: "cod0 11 = 10"
  using id0_le_id1 local.csr0.rdual.dom_iso local.dual_order.antisym by fastforce

lemma c0_id0: "cod1 10 = 10"
  using c1d0 d0_id1 by blast

lemma comm_d0d1: "dom0 (dom1 x) = dom1 (dom0 x)"
proof-
  have "dom0 (dom1 x) = dom0 (dom1 (dom0 x ⋅0 x))"
    by simp
  also have "… ≤ dom0 (dom1 (dom0 x) ⋅0 dom1 x)"
    using local.d1_hom local.dsr0.d_iso by blast
  also have "… = dom0 (dom0 x ⋅0 dom1 x)"
    by simp
  also have "… = dom0 x ⋅0 dom0 (dom1 x)"
    by simp
  also have "… = dom1 (dom0 x) ⋅0 dom0 (dom1 x)"
    by simp
  also have "… ≤ dom1 (dom0 x) ⋅0 10"
    using d0.mult_isol local.d0_subid by blast
  finally have a: "dom0 (dom1 x) ≤ dom1 (dom0 x)"
    by simp
  have "dom1 (dom0 x) = dom0 x"
    by simp
  also have "… = dom0 (dom1 x ⋅1 x)"
    by simp
  also have "… ≤ dom0 (dom1 x) ⋅1 dom0 x"
    using local.d0_hom by blast 
  also have "… ≤ dom0 (dom1 x) ⋅1 10"
    by (simp add: d1.mult_isol)
  also have "… ≤ dom0 (dom1 x) ⋅1 11"
    using d1.mult_isol local.id0_le_id1 by presburger
  finally have "dom1 (dom0 x) ≤ dom0 (dom1 x)"
    by simp
  thus ?thesis
    using a by auto
qed

lemma comm_d0c1: "dom0 (cod1 x) = cod1 (dom0 x)"
proof-
 have "dom0 (cod1 x) = dom0 (cod1 (dom0 x ⋅0 x))"
    by simp
  also have "… ≤ dom0 (cod1 (dom0 x) ⋅0 cod1 x)"
    using local.c1_hom local.dsr0.d_iso by blast
  also have "… = dom0 (dom0 x ⋅0 cod1 x)"
    by simp
  also have "… = dom0 x ⋅0 dom0 (cod1 x)"
    by simp
  also have "… = cod1 (dom0 x) ⋅0 dom0 (cod1 x)"
    by simp
  also have "… ≤ cod1 (dom0 x) ⋅0 10"
    using d0.mult_isol local.d0_subid by blast
  finally have a: "dom0 (cod1 x) ≤ cod1 (dom0 x)"
    by simp
  have "cod1 (dom0 x) = dom0 x"
    by simp
  also have "… = dom0 (x ⋅1 cod1 x)"
    by simp
  also have "… ≤ dom0 x ⋅1 dom0 (cod1 x)"
    using local.d0_hom by blast
  also have  "… ≤ 10 ⋅1 dom0 (cod1 x)"
    by (simp add: d1.mult_isor)
  also have  "… ≤ 11 ⋅1 dom0 (cod1 x)"
    using d1.mult_isor local.id0_le_id1 by blast
  finally have "cod1 (dom0 x) ≤ dom0 (cod1 x)"
    by simp
  thus ?thesis
    using a by auto
qed

lemma comm_c0c1: "cod0 (cod1 x) = cod1 (cod0 x)"
  by (metis c1c0 local.csr0.rdual.dom_llp local.csr0.rdual.dom_ord local.csr0.rdual.dsg1 local.csr0.rdual.dsg4 local.csr1.rdual.dom_llp local.csr1.rdual.dsg1 local.tgsdual.d0_hom local.tgsdual.d1_hom)

lemma comm_c0d1: "cod0 (dom1 x) = dom1 (cod0 x)"
  by (metis d1c0 local.c0_hom local.csr0.rdual.dom_subid_aux2 local.csr0.rdual.domain_1'' local.csr0.rdual.domain_invol local.csr0.rdual.dsg1 local.d1_hom local.dsr1.dom_subid_aux2 local.dsr1.dom_subid_aux2'' local.dsr1.dsg1 local.dual_order.antisym)

text ‹We prove further lemmas that are not related to the globular structure.›

lemma d0_comp1_idem [simp]: "dom0 x ⋅1 dom0 x = dom0 x"
proof-
  have "dom0 x ⋅1 dom0 x = dom1 (dom0 x) ⋅1 dom1 (dom0 x)"
    by simp
  also have "… = dom1 (dom0 x)"
    using local.dsr1.dom_el_idem by blast
  also have "… =  dom0 x"
    by simp
  finally show ?thesis.
qed

lemma cod0_comp1_idem: "cod0 x ⋅1 cod0 x = cod0 x"
  by (metis d1c0 local.dsr1.dsg1)

lemma  dom01_loc [simp]: "dom0 (x ⋅1 dom1 y) = dom0 (x ⋅1 y)"
proof-
  have "dom0 (x ⋅1 y) = dom0 (dom1 (x ⋅1 y))"
    by (simp add: local.comm_d0d1)
  also have "… = dom0 (dom1 (x ⋅1 dom1 y))"
    by simp
  also have "… = dom0 (x ⋅1 dom1 y)"
    using local.comm_d0d1 local.d1d0 by presburger
  finally show ?thesis..
qed

lemma cod01_locl: "cod0 (cod1 x ⋅1 y) = cod0 (x ⋅1 y)"
  by (metis c1c0 comm_c0c1 local.cod1_local)

lemma dom01_exp [simp]: "dom0 (cod1 x ⋅1 y) = dom0 (x ⋅1 y)"
  by (metis local.c1d0 local.cod1_local local.comm_d0c1)

lemma cod01_exo: "cod0 (x ⋅1 dom1 y) = cod0 (x ⋅1 y)"
  by (metis comm_c0d1 d1c0 local.d1_local)

lemma dom01_loc_var [simp]: "dom0 (x ⋅0 dom1 y) = dom0 (x ⋅0 y)"
proof-
  have "dom0 (x ⋅0 y) = dom0 (x ⋅0 dom0 y)"
    by simp
  also have "… = dom0 (x ⋅0 dom1 (dom0 y))"
    by simp
  also have "… = dom0 (x ⋅0 dom0 (dom1 y))"
    by (simp add: local.comm_d0d1)
  also have "… = dom0 (x ⋅0 dom1 y)"
    by simp
    finally show ?thesis..
  qed

lemma cod01_loc_var: "cod0 (cod1 x ⋅0 y) = cod0 (x ⋅0 y)"
  by (metis c1c0 comm_c0c1 local.cod0_local)

lemma dom0cod1_exp: "dom0 (x ⋅0 y) ≤ dom0 (cod1 x ⋅0 y)"
proof-
  have "dom0 (x ⋅0 y) = dom0 (cod1 (x ⋅0 y))"
    by (simp add: local.comm_d0c1)
  also have "… ≤ dom0 (cod1 x ⋅0 cod1 y)"
    by (simp add: local.c1_hom local.dsr0.d_iso)
  also have "… = dom0 (cod1 x ⋅0 dom0 (cod1 y))"
    by simp
  also have "… = dom0 (cod1 x ⋅0 dom0 y)"
    by (simp add: local.comm_d0c1)
  also have "… = dom0 (cod1 x ⋅0 y)"
    by simp
  finally show ?thesis.
qed

lemma cod0dom1_exp: " cod0 (x ⋅0 y) ≤ cod0 (x ⋅0 dom1 y)"
  by (metis comm_c0d1 d1c0 local.cod0_local local.csr0.rdual.dom_iso local.d1_hom)

lemma (in two_semiring) d0_comp1: "dom0 x ⋅0 (y ⋅1 z) ≤ (dom0 x ⋅0 y) ⋅1 (dom0 x ⋅0 z)"
proof-
  have "dom0 x ⋅0 (y ⋅1 z) = (dom0 x ⋅1 dom0 x) ⋅0 (y ⋅1 z)"
    by simp
  also have "… ≤ (dom0 x ⋅0 y) ⋅1 (dom0 x ⋅0 z)"
    using local.interchange by presburger
  finally show ?thesis.
qed

lemma d1_comp1: "dom1 x ⋅0 (y ⋅1 z) ≤ (dom1 x ⋅0 y) ⋅1 (dom1 x ⋅0 z)"
  by (metis local.dsr1.dom_el_idem local.tgsdual.interchange)

lemma d01_export: "dom0 (dom1 x ⋅1 y) ≤ dom0 x ⋅1 dom0 y"
proof-
  have "dom0 (dom1 x ⋅1 y) ≤ dom0 (dom1 x) ⋅1 dom0 y"
    by (simp add: local.d0_hom)
  also have "… = dom0 x ⋅1 dom0 y"
    by (simp add: local.comm_d0d1)
  finally show ?thesis.
qed

lemma cod01_export: "cod0 (x ⋅1 cod1 y) ≤ cod0 x ⋅1 cod0 y"
  by (metis local.c0_hom local.c1c0 local.comm_c0c1)

lemma d10_export [simp]: "dom1 (dom0 x ⋅1 y) = dom0 x ⋅1 dom1 y"
  by (metis local.d1d0 local.dsr1.dsg3)

lemma cod10_export: "cod1 (x ⋅1 cod0 y) = cod1 x ⋅1 cod0 y"
  by (metis local.c1c0 local.csr1.rdual.dsg3)

lemma d0_comp10: "dom0 x ⋅1 dom0 y = dom0 x ⋅0 dom0 y"
proof (rule order.antisym)
  have "dom0 x ⋅1 dom0 y ≤ dom0 (dom0 x ⋅1 dom0 y) ⋅0 (dom0 x ⋅1 dom0 y)"
    by simp
  also have "… ≤ (dom0 (dom0 x) ⋅1 dom0 (dom0 y)) ⋅0 (dom0 x ⋅1 dom0 y)"
    using d0.mult_isor local.tgsdual.c0_hom by blast
  also have "… ≤ (dom0 x ⋅1 10) ⋅0 (10 ⋅1 dom0 y)"
    by (simp add: local.d0.mult_isol_var local.d1.mult_isol_var)
  also have "… ≤ (dom0 x ⋅1 11) ⋅0 (11 ⋅1 dom0 y)"
    using local.d0.mult_isol_var local.dd1.d1.mult_isol local.dd1.d1.mult_isor local.id0_le_id1 by presburger
  also have "… ≤ dom0 x ⋅0 dom0 y"
    by simp
  finally show "dom0 x ⋅1 dom0 y ≤ dom0 x ⋅0 dom0 y".
next
  have "dom0 x ⋅0 dom0 y = (dom0 x ⋅1 dom0 x) ⋅0 (dom0 y ⋅1 dom0 y)"
    by simp
  also have "… ≤ (dom0 x ⋅0 dom0 y) ⋅1 (dom0 x ⋅0 dom0 y)"
    using local.interchange by blast
  also have "… ≤ (dom0 x ⋅0 10) ⋅1 (10 ⋅0 dom0 y)"
    by (simp add: local.d1.mult_isol_var local.dsr0.dom_subid_aux2 local.dsr0.dom_subid_aux2'')
  also have "… = dom0 x ⋅1 dom0 y"
    by simp
  finally show "dom0 x ⋅0 dom0 y ≤ dom0 x ⋅1 dom0 y".
qed

lemma dom_exchange_strong: "(dom0 w ⋅1 dom0 x) ⋅0 (dom0 y ⋅1 dom0 z) = (dom0 w ⋅0 dom0 y) ⋅1 (dom0 x ⋅0 dom0 z)"
proof-
  have  "(dom0 w ⋅1 dom0 x) ⋅0 (dom0 y ⋅1 dom0 z) = (dom0 w ⋅0 dom0 x) ⋅0 (dom0 y ⋅0 dom0 z)"
    by (simp add: local.d0_comp10)
  also have "… = (dom0 w ⋅0 dom0 y) ⋅0 (dom0 x ⋅0 dom0 z)"
    by (metis local.dd0.mm0.mult_assoc local.dsr0.dsg4)
  also have  "… = dom0 (dom0 w ⋅0 dom0 y) ⋅0 dom0 (dom0 x ⋅0 dom0 z)"
    by simp
  also have  "… = dom0 (dom0 w ⋅0 dom0 y) ⋅1 dom0 (dom0 x ⋅0 dom0 z)"
    using local.d0_comp10 by presburger
  also have  "… = (dom0 w ⋅0 dom0 y) ⋅1 (dom0 x ⋅0 dom0 z)"
    by simp
  finally show ?thesis.
qed

end

context strong_two_semiring
begin

lemma id1_comp0: "11 ⋅0 11 ≤ 11"
proof-
  have "11 ⋅0 11 = dom1 11 ⋅0 dom1 11"
    by simp
  also have "… = dom1 (11 ⋅0 11)"
    by simp
  also have "… ≤ 11"
    using local.d1_subid by blast
  finally show ?thesis.
qed

lemma id1_comp0_eq [simp]: "11 ⋅0 11 = 11"
proof-
  have "11 = 11 ⋅0 10"
    by simp
  also have "… = (11 ⋅1 11) ⋅0 (10 ⋅1 11)"
    by simp
  also have "… ≤ (11 ⋅0 10) ⋅1 (11 ⋅0 11)"
    using local.interchange by presburger
  also have "… = 11 ⋅1 (11 ⋅0 11)"
    by simp
  also have "… = 11 ⋅0 11"
    by simp
  finally have "11 ≤ 11 ⋅0 11".
  thus ?thesis
    by (simp add: local.antisym_conv local.id1_comp0)
qed

lemma  "10 = 11"
  (*nitpick  *)
  oops

lemma dom0cod1_exp: "dom0 (x ⋅0 y) = dom0 (cod1 x ⋅0 y)"
proof-
  have "dom0 (x ⋅0 y) = dom0 (cod1 (x ⋅0 y))"
    using local.c1d0 local.comm_d0c1 by presburger
  also have "… = dom0 (cod1 x ⋅0 cod1 y)"
    by (simp add: local.c1_hom local.dsr0.d_iso)
  also have "… = dom0 (cod1 x ⋅0 dom0 (cod1 y))"
    by simp
  also have "… = dom0 (cod1 x ⋅0 dom0 y)"
    by (simp add: local.comm_d0c1)
  also have "… = dom0 (cod1 x ⋅0 y)"
    by simp
  finally show ?thesis.
qed

lemma cod0dom1_exp: "cod0 (x ⋅0 dom1 y) = cod0 (x ⋅0 y)"
  by (metis local.comm_c0d1 local.d1c0 local.ds0dual.d0_local local.stgsdual.c1_strong_hom)

end

text ‹The following laws are diamond laws. It remains to define diamonds for them.›

context two_semiring
begin

lemma fdia0fdia1_prop: "dom0 (y ⋅0 dom1 (x ⋅1 z)) = dom0 (y ⋅0 (x ⋅1 z))"
  by simp

lemma bdia0fdia1_prop [simp]: "cod0 (dom1 (x ⋅1 z) ⋅0 y) = cod0 ((x ⋅1 z) ⋅0 y)"
  by (metis local.comm_c0d1 local.d1c0 local.ds0dual.d0_local)

lemma fdia0bdia1_prop: "dom0 (y ⋅0 cod1 (x ⋅1 z)) = dom0 (y ⋅0 (x ⋅1 z))"
  by (metis local.c1d0 local.comm_d0c1 local.d0_local)

lemma bdia0bdia1_prop: "cod0 (cod1 (x ⋅1 z) ⋅0 y) = cod0 ((x ⋅1 z) ⋅0 y)"
  by simp

lemma fdia0fdia1_prop2: "dom0 (y ⋅0 dom1 (x ⋅1 z)) ≤ dom0 (y ⋅0 (dom0 x ⋅1 dom0 z))"
proof-
  have "dom0 (y ⋅0 dom1 (x ⋅1 z)) = dom0 (y ⋅0 dom0 (x ⋅1 z))"
    by simp
  also have "… ≤ dom0 (y ⋅0 (dom0 x ⋅1 dom0 z))"
    using d0.mult_isol local.dsr0.d_iso local.tgsdual.c0_hom by presburger
  finally show ?thesis.
qed

lemma fdia00_prop2: "dom0 (y ⋅0 dom0 (x ⋅1 z)) ≤ dom0 (y ⋅0 (dom0 x ⋅1 dom0 z))"
  using local.fdia0fdia1_prop2 by auto
 
lemma bdia0dom1_prop2: "cod0 (dom1 (x ⋅1 z) ⋅0 y) ≤ cod0 ((cod0 x ⋅1 cod0 z) ⋅0 y)"
  using local.c0_hom local.csr0.rdual.fd_def local.csr0.rdual.fd_iso1 local.tgsdual.d0_comp10 by auto

lemma bdia0dom0_prop2: "cod0 (dom0 (x ⋅1 z) ⋅0 y) ≤ cod0 ((dom0 x ⋅1 dom0 z) ⋅0 y)"
  by (simp add: local.csr0.rdual.dom_iso local.dd0.d0.mult_isol local.tgsdual.c0_hom)

lemma fdia0bdia1_prop_2: "dom0 (y ⋅0 cod1 (z ⋅1 x)) ≤ dom0 (y ⋅0 (dom0 x ⋅1 dom0 z))"
  by (metis fdia00_prop2 local.c1d0 local.csr1.rdual.dsg1 local.csr1.rdual.dsg4 local.dom01_exp local.msr0dual.cod0_local)

lemma fdia0bdiao_prop2: "dom0 (y ⋅0 cod0 (z ⋅1 x)) ≤ dom0 (y ⋅0 (cod0 z ⋅1 cod0 x))"
  by (simp add: local.c0_hom local.dd0.d0.mult_isor local.dsr0.dom_iso)

lemma bdia0bdia1_prop2: "cod0 (cod1 (z ⋅1 x) ⋅0 y) ≤ cod0 ((cod0 x ⋅1 cod0 z) ⋅0 y)"
  using bdia0dom1_prop2 local.csr0.rdual.dsg4 local.tgsdual.d0_comp10 by fastforce

lemma bdia0bdia0_prop2: "cod0 (cod0 (x ⋅1 z) ⋅0 y) ≤ cod0 ((cod0 x ⋅1 cod0 z) ⋅0 y)"
  using bdia0dom1_prop2 by force

lemma fdia1fdia0_prop3 [simp]: "dom1 (x ⋅1 dom0 (y ⋅0 z)) ≤ dom1 (x ⋅1 dom0 (dom1 y ⋅0 z))"
  by (metis d1.mult_isol local.comm_d0d1 local.d1_hom local.d1d0 local.dsr0.d_iso local.dsr1.d_iso local.tgsdual.cod01_loc_var)

lemma fdia1bdia0_prop3 [simp]: "dom1 (x ⋅1 cod0 (z ⋅0 y)) ≤ dom1 (x ⋅1 cod0 (z ⋅0 dom1 y))"
  by (simp add: d1.mult_isol local.dsr1.d_iso local.tgsdual.dom0cod1_exp)

lemma bdia1fdia0_prop3: "cod1 (dom0 (y ⋅0 z) ⋅1 x) ≤ cod1 (dom0 (cod1 y ⋅0 z) ⋅1 x)"
  by (simp add: local.csr1.rdual.dom_iso local.dd1.d1.mult_isol local.tgsdual.cod0dom1_exp)

lemma bdia1bdia0_prop3: "cod1 (cod0 (z ⋅0 y) ⋅1 x) ≤ cod1 (cod0 (z ⋅0 cod1 y) ⋅1 x)"
  by (metis local.c1_hom local.c1c0 local.comm_c0c1 local.csr0.rdual.dom_iso local.csr1.rdual.dom_iso local.dd1.d1.mult_isol local.tgsdual.dom01_loc_var)

end

context strong_two_semiring
begin

lemma fdia1fdia0_prop3 [simp]: "dom1 (x ⋅1 dom0 (dom1 y ⋅0 z)) = dom1 (x ⋅1 dom0 (y ⋅0 z))"
  by (metis local.comm_d0d1 local.d1_strong_hom local.d1d0 local.dom01_loc_var)

lemma fdia1bdia0_prop3 [simp]: "dom1 (x ⋅1 cod0 (z ⋅0 dom1 y)) = dom1 (x ⋅1 cod0 (z ⋅0 y))"
  by (simp add: local.cod0dom1_exp)

lemma bdia1fdia0_prop3: "cod1 (dom0 (cod1 y ⋅0 z) ⋅1 x) = cod1 (dom0 (y ⋅0 z) ⋅1 x)"
  by (simp add: local.stgsdual.cod0dom1_exp)

lemma bdia1bdia0_prop3: " cod1 (cod0 (z ⋅0 cod1 y) ⋅1 x) = cod1 (cod0 (z ⋅0 y) ⋅1 x)"
  by (metis local.c1_strong_hom local.c1c0 local.cod01_loc_var local.comm_c0c1)

lemma fdia0fdia1_prop4: "dom0 z ⋅0 dom1 (x ⋅1 y) ≤ dom1 ((dom0 z ⋅0 x) ⋅1 (dom0 z ⋅0 y))"
 proof-
  have  "dom0 z ⋅0 dom1 (x ⋅1 y) = dom1 (dom0 z) ⋅0 dom1 (x ⋅1 y)"
    by simp
  also have "… = dom1 (dom0 z ⋅0 (x ⋅1 y))"
    by simp
  also have "… ≤ dom1 ((dom0 z ⋅0 x) ⋅1 (dom0 z  ⋅0 y))"
    using local.d0_comp1 local.dsr1.d_iso by presburger
  finally show ?thesis.
qed

lemma fdia0bdia1_prop4: "dom0 z ⋅0 cod1 (y ⋅1 x) ≤ cod1 ((dom0 z ⋅0 y) ⋅1 (dom0 z ⋅0 x))"
  by (metis local.c1d0 local.csr1.rdual.dom_iso local.d0_comp1 local.stgsdual.d1_strong_hom)

lemma fdia1fdia1_prop4: "dom1 (x ⋅1 y) ⋅0 dom0 z ≤ dom1 ((x ⋅0 dom0 z) ⋅1 (y ⋅0 dom0 z))"
  by (metis local.d0_comp1_idem local.d1_strong_hom local.d1d0 local.dsr1.d_iso local.tgsdual.interchange)

lemma bdia1bdia1_prop4:  "cod1 (y ⋅1 x) ⋅0 dom0 z ≤ cod1 ((y ⋅0 dom0 z) ⋅1 (x ⋅0 dom0 z))"
  by (metis local.c1d0 local.csr1.rdual.dom_iso local.stgsdual.d1_strong_hom local.tgsdual.d1_comp1)

end

subsection ‹Globular 2-Kleene algebras›

class two_kleene_algebra = two_semiring + kleene_algebra0 + kleene_algebra1 

class strong_two_kleene_algebra = strong_two_semiring + kleene_algebra0 + kleene_algebra1 

lemma (in strong_two_kleene_algebra) "star1 x ⋅0 star1 y ≤ star0 (x ⋅1 y)"
  (*nitpick *)
  oops

lemma (in strong_two_kleene_algebra) "star1 x ⋅0 star1 y ≤ star1 (x ⋅1 y)"
  (*nitpick *)
  oops

context two_kleene_algebra
begin

lemma interchange_var1: "(x ⋅1 x) ⋅0 (y ⋅1 y) ⋅0 (z ⋅1 z) ≤ (x ⋅0 y ⋅0 z) ⋅1 (x ⋅0 y ⋅0 z)"
  by (meson local.dd0.d0.mult_isol local.dual_order.trans local.tgsdual.interchange)

lemma interchange_var2: "(x ⋅1 y) ⋅0 (x ⋅1 y) ⋅0 (x ⋅1 y) ≤ (x ⋅0 x ⋅0 x) ⋅1 (y ⋅0 y ⋅0 y)"
  by (meson local.dd0.d0.mult_isol local.dual_order.trans local.tgsdual.interchange)

lemma star0_comp1: "star0 (x ⋅1 y) ≤ star0 x ⋅1 star0 y"
proof-
  have a: "10 ≤ star0 x ⋅1 star0 y"
    by (metis local.d1.mult_isol_var local.id0_comp1_eq local.k0.star_ref)
  have "(x ⋅1 y) ⋅0 (star0 x ⋅1 star0 y) ≤ (x ⋅0 star0 x ) ⋅1 (y ⋅0 star0 y)"
    by (simp add: local.interchange)
  also have "… ≤ star0 x ⋅1 star0 y"
    by (simp add: local.dd1.d1.mult_isol_var)
  finally have "(x ⋅1 y) ⋅0 (star0 x ⋅1 star0 y) ≤ star0 x ⋅1 star0 y".
  hence "10 + (x ⋅1 y) ⋅0 (star0 x ⋅1 star0 y) ≤ star0 x ⋅1 star0 y"
    by (simp add: a)
  thus ?thesis
    using local.star0_inductl by force
qed

end

context strong_two_kleene_algebra
begin

lemma " star1 (x ⋅1 y) ≤ star1 x ⋅0 star1 y"
  (*nitpick *)
  oops

lemma "star1 x ⋅0 star1 y ≤ star1 (x ⋅0 y)"
   (*nitpick *)
  oops

lemma "star1 (x ⋅0 y) ≤ star1 x ⋅0 star1 y"
   (*nitpick *)
  oops

lemma "star0 x ⋅1 star0 y ≤ star0 (x ⋅0 y)"
   (*nitpick *)
  oops

lemma " star0 (x ⋅0 y) ≤ star0 x ⋅1 star0 y"
   (*nitpick *)
  oops

lemma "star0 x ⋅1 star0 y ≤ star0 (x ⋅1 y)"
   (*nitpick *)
  oops

lemma (in strong_two_kleene_algebra) "dom0 x ⋅0 star1 y ≤ star1 (dom0 x ⋅0 y)" 
  oops (* no proof no counterexample *)

end
 
class two_quantale_lmcs = modal_semiring0 + modal_semiring1 +
  assumes interchange: "(w ⋅1 x) ⋅0 (y ⋅1 z) ≤ (w ⋅0 y) ⋅1 (x ⋅0 z)"
  and d1_hom: "dom1 (x ⋅0 y) = dom1 x ⋅0 dom1 y"
  and c1_hom: "cod1 (x ⋅0 y) = cod1 x ⋅0 cod1 y"
  and d1d0 [simp]: "dom1 (dom0 x) = dom0 x"
 and c1d0 [simp]: "cod1 (dom0 x) = dom0 x"
 and d1c0 [simp]: "dom1 (cod0 x) = cod0 x"
 and c1c0 [simp]: "cod1 (cod0 x) = cod0 x"
  and d0d1 [simp]: "dom0 (dom1 x) = dom0 x"
 and c0d1 [simp]: "cod0 (dom1 x) = dom0 x"
 and d0c1 [simp]: "dom0 (cod1 x) = cod0 x"
 and c0c1 [simp]: "cod0 (cod1 x) = cod0 x"

begin

lemma "dom0 (x ⋅1 y) ≤ dom0 x ⋅1 dom0 y"
  (* no proof no counterexample *)
  oops

lemma  "cod0 (x ⋅1 y) ≤ cod0 x ⋅1 cod0 y"
  (* no proof no counterexample *)
  oops

end

end