(*  Title:      HOL/HOLCF/IOA/CompoExecs.thy
    Author:     Olaf Müller
*)

section ‹Compositionality on Execution level›

theory CompoExecs
imports Traces
begin

definition ProjA2 :: "('a, 's × 't) pairs → ('a, 's) pairs"
  where "ProjA2 = Map (λx. (fst x, fst (snd x)))"

definition ProjA :: "('a, 's × 't) execution → ('a, 's) execution"
  where "ProjA ex = (fst (fst ex), ProjA2 ⋅ (snd ex))"

definition ProjB2 :: "('a, 's × 't) pairs → ('a, 't) pairs"
  where "ProjB2 = Map (λx. (fst x, snd (snd x)))"

definition ProjB :: "('a, 's × 't) execution → ('a, 't) execution"
  where "ProjB ex = (snd (fst ex), ProjB2 ⋅ (snd ex))"

definition Filter_ex2 :: "'a signature → ('a, 's) pairs → ('a, 's) pairs"
  where "Filter_ex2 sig = Filter (λx. fst x ∈ actions sig)"

definition Filter_ex :: "'a signature → ('a, 's) execution → ('a, 's) execution"
  where "Filter_ex sig ex = (fst ex, Filter_ex2 sig ⋅ (snd ex))"

definition stutter2 :: "'a signature → ('a, 's) pairs → ('s → tr)"
  where "stutter2 sig =
    (fix ⋅
      (LAM h ex.
        (λs.
          case ex of
            nil → TT
          | x ## xs →
              (flift1
                (λp.
                  (If Def (fst p ∉ actions sig) then Def (s = snd p) else TT)
                  andalso (h⋅xs) (snd p)) ⋅ x))))"

definition stutter :: "'a signature → ('a, 's) execution → bool"
  where "stutter sig ex ⟷ (stutter2 sig ⋅ (snd ex)) (fst ex) ≠ FF"

definition par_execs ::
  "('a, 's) execution_module → ('a, 't) execution_module → ('a, 's × 't) execution_module"
  where "par_execs ExecsA ExecsB =
    (let
      exA = fst ExecsA; sigA = snd ExecsA;
      exB = fst ExecsB; sigB = snd ExecsB
     in
       ({ex. Filter_ex sigA (ProjA ex) ∈ exA} ∩
        {ex. Filter_ex sigB (ProjB ex) ∈ exB} ∩
        {ex. stutter sigA (ProjA ex)} ∩
        {ex. stutter sigB (ProjB ex)} ∩
        {ex. Forall (λx. fst x ∈ actions sigA ∪ actions sigB) (snd ex)},
        asig_comp sigA sigB))"


lemmas [simp del] = split_paired_All


section ‹Recursive equations of operators›

subsection ‹‹ProjA2››

lemma ProjA2_UU: "ProjA2 ⋅ UU = UU"
  by (simp add: ProjA2_def)

lemma ProjA2_nil: "ProjA2 ⋅ nil = nil"
  by (simp add: ProjA2_def)

lemma ProjA2_cons: "ProjA2 ⋅ ((a, t) ↝ xs) = (a, fst t) ↝ ProjA2 ⋅ xs"
  by (simp add: ProjA2_def)


subsection ‹‹ProjB2››

lemma ProjB2_UU: "ProjB2 ⋅ UU = UU"
  by (simp add: ProjB2_def)

lemma ProjB2_nil: "ProjB2 ⋅ nil = nil"
  by (simp add: ProjB2_def)

lemma ProjB2_cons: "ProjB2 ⋅ ((a, t) ↝ xs) = (a, snd t) ↝ ProjB2 ⋅ xs"
  by (simp add: ProjB2_def)


subsection ‹‹Filter_ex2››

lemma Filter_ex2_UU: "Filter_ex2 sig ⋅ UU = UU"
  by (simp add: Filter_ex2_def)

lemma Filter_ex2_nil: "Filter_ex2 sig ⋅ nil = nil"
  by (simp add: Filter_ex2_def)

lemma Filter_ex2_cons:
  "Filter_ex2 sig ⋅ (at ↝ xs) =
    (if fst at ∈ actions sig
     then at ↝ (Filter_ex2 sig ⋅ xs)
     else Filter_ex2 sig ⋅ xs)"
  by (simp add: Filter_ex2_def)


subsection ‹‹stutter2››

lemma stutter2_unfold:
  "stutter2 sig =
    (LAM ex.
      (λs.
        case ex of
          nil → TT
        | x ## xs →
            (flift1
              (λp.
                (If Def (fst p ∉ actions sig) then Def (s= snd p) else TT)
                andalso (stutter2 sig⋅xs) (snd p)) ⋅ x)))"
  apply (rule trans)
  apply (rule fix_eq2)
  apply (simp only: stutter2_def)
  apply (rule beta_cfun)
  apply (simp add: flift1_def)
  done

lemma stutter2_UU: "(stutter2 sig ⋅ UU) s = UU"
  apply (subst stutter2_unfold)
  apply simp
  done

lemma stutter2_nil: "(stutter2 sig ⋅ nil) s = TT"
  apply (subst stutter2_unfold)
  apply simp
  done

lemma stutter2_cons:
  "(stutter2 sig ⋅ (at ↝ xs)) s =
    ((if fst at ∉ actions sig then Def (s = snd at) else TT)
      andalso (stutter2 sig ⋅ xs) (snd at))"
  apply (rule trans)
  apply (subst stutter2_unfold)
  apply (simp add: Consq_def flift1_def If_and_if)
  apply simp
  done

declare stutter2_UU [simp] stutter2_nil [simp] stutter2_cons [simp]


subsection ‹‹stutter››

lemma stutter_UU: "stutter sig (s, UU)"
  by (simp add: stutter_def)

lemma stutter_nil: "stutter sig (s, nil)"
  by (simp add: stutter_def)

lemma stutter_cons:
  "stutter sig (s, (a, t) ↝ ex) ⟷ (a ∉ actions sig ⟶ (s = t)) ∧ stutter sig (t, ex)"
  by (simp add: stutter_def)

declare stutter2_UU [simp del] stutter2_nil [simp del] stutter2_cons [simp del]

lemmas compoex_simps = ProjA2_UU ProjA2_nil ProjA2_cons
  ProjB2_UU ProjB2_nil ProjB2_cons
  Filter_ex2_UU Filter_ex2_nil Filter_ex2_cons
  stutter_UU stutter_nil stutter_cons

declare compoex_simps [simp]


section ‹Compositionality on execution level›

lemma lemma_1_1a:
  ― ‹‹is_ex_fr› propagates from ‹A ∥ B› to projections ‹A› and ‹B››
  "∀s. is_exec_frag (A ∥ B) (s, xs) ⟶
    is_exec_frag A (fst s, Filter_ex2 (asig_of A) ⋅ (ProjA2 ⋅ xs)) ∧
    is_exec_frag B (snd s, Filter_ex2 (asig_of B) ⋅ (ProjB2 ⋅ xs))"
  apply (pair_induct xs simp: is_exec_frag_def)
  text ‹main case›
  apply (auto simp add: trans_of_defs2)
  done

lemma lemma_1_1b:
  ― ‹‹is_ex_fr (A ∥ B)› implies stuttering on projections›
  "∀s. is_exec_frag (A ∥ B) (s, xs) ⟶
    stutter (asig_of A) (fst s, ProjA2 ⋅ xs) ∧
    stutter (asig_of B) (snd s, ProjB2 ⋅ xs)"
  apply (pair_induct xs simp: stutter_def is_exec_frag_def)
  text ‹main case›
  apply (auto simp add: trans_of_defs2)
  done

lemma lemma_1_1c:
  ― ‹Executions of ‹A ∥ B› have only ‹A›- or ‹B›-actions›
  "∀s. is_exec_frag (A ∥ B) (s, xs) ⟶ Forall (λx. fst x ∈ act (A ∥ B)) xs"
  apply (pair_induct xs simp: Forall_def sforall_def is_exec_frag_def)
  text ‹main case›
  apply auto
  apply (simp add: trans_of_defs2 actions_asig_comp asig_of_par)
  done

lemma lemma_1_2:
  ― ‹‹ex A›, ‹exB›, stuttering and forall ‹a ∈ A ∥ B› implies ‹ex (A ∥ B)››
  "∀s.
    is_exec_frag A (fst s, Filter_ex2 (asig_of A) ⋅ (ProjA2 ⋅ xs)) ∧
    is_exec_frag B (snd s, Filter_ex2 (asig_of B) ⋅ (ProjB2 ⋅ xs)) ∧
    stutter (asig_of A) (fst s, ProjA2 ⋅ xs) ∧
    stutter (asig_of B) (snd s, ProjB2 ⋅ xs) ∧
    Forall (λx. fst x ∈ act (A ∥ B)) xs ⟶
    is_exec_frag (A ∥ B) (s, xs)"
  apply (pair_induct xs simp: Forall_def sforall_def is_exec_frag_def stutter_def)
  apply (auto simp add: trans_of_defs1 actions_asig_comp asig_of_par)
  done

theorem compositionality_ex:
  "ex ∈ executions (A ∥ B) ⟷
    Filter_ex (asig_of A) (ProjA ex) ∈ executions A ∧
    Filter_ex (asig_of B) (ProjB ex) ∈ executions B ∧
    stutter (asig_of A) (ProjA ex) ∧
    stutter (asig_of B) (ProjB ex) ∧
    Forall (λx. fst x ∈ act (A ∥ B)) (snd ex)"
  apply (simp add: executions_def ProjB_def Filter_ex_def ProjA_def starts_of_par)
  apply (pair ex)
  apply (rule iffI)
  text ‹‹==>››
  apply (erule conjE)+
  apply (simp add: lemma_1_1a lemma_1_1b lemma_1_1c)
  text ‹‹<==››
  apply (erule conjE)+
  apply (simp add: lemma_1_2)
  done

theorem compositionality_ex_modules: "Execs (A ∥ B) = par_execs (Execs A) (Execs B)"
  apply (unfold Execs_def par_execs_def)
  apply (simp add: asig_of_par)
  apply (rule set_eqI)
  apply (simp add: compositionality_ex actions_of_par)
  done

end