Theory Deadlock

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

section ‹Deadlock freedom of I/O Automata›

theory Deadlock
imports RefCorrectness CompoScheds
begin

text ‹Input actions may always be added to a schedule.›

lemma scheds_input_enabled:
  "Filter (λx. x ∈ act A) ⋅ sch ∈ schedules A ⟹ a ∈ inp A ⟹ input_enabled A ⟹ Finite sch
    ⟹ Filter (λx. x ∈ act A) ⋅ sch @@ a ↝ nil ∈ schedules A"
  apply (simp add: schedules_def has_schedule_def)
  apply auto
  apply (frule inp_is_act)
  apply (simp add: executions_def)
  apply (pair ex)
  apply (rename_tac s ex)
  apply (subgoal_tac "Finite ex")
  prefer 2
  apply (simp add: filter_act_def)
  defer
  apply (rule_tac [2] Map2Finite [THEN iffD1])
  apply (rule_tac [2] t = "Map fst ⋅ ex" in subst)
  prefer 2
  apply assumption
  apply (erule_tac [2] FiniteFilter)
  text ‹subgoal 1›
  apply (frule exists_laststate)
  apply (erule allE)
  apply (erule exE)
  text ‹using input-enabledness›
  apply (simp add: input_enabled_def)
  apply (erule conjE)+
  apply (erule_tac x = "a" in allE)
  apply simp
  apply (erule_tac x = "u" in allE)
  apply (erule exE)
  text ‹instantiate execution›
  apply (rule_tac x = " (s, ex @@ (a, s2) ↝ nil) " in exI)
  apply (simp add: filter_act_def MapConc)
  apply (erule_tac t = "u" in lemma_2_1)
  apply simp
  apply (rule sym)
  apply assumption
  done

text ‹
  Deadlock freedom: component B cannot block an out or int action of component
  A in every schedule.

  Needs compositionality on schedule level, input-enabledness, compatibility
  and distributivity of ‹is_exec_frag› over ‹@@›.
›

lemma IOA_deadlock_free:
  assumes "a ∈ local A"
    and "Finite sch"
    and "sch ∈ schedules (A ∥ B)"
    and "Filter (λx. x ∈ act A) ⋅ (sch @@ a ↝ nil) ∈ schedules A"
    and "compatible A B"
    and "input_enabled B"
  shows "(sch @@ a ↝ nil) ∈ schedules (A ∥ B)"
  supply if_split [split del]
  apply (insert assms)
  apply (simp add: compositionality_sch locals_def)
  apply (rule conjI)
  text ‹‹a ∈ act (A ∥ B)››
  prefer 2
  apply (simp add: actions_of_par)
  apply (blast dest: int_is_act out_is_act)

  text ‹‹Filter B (sch @@ [a]) ∈ schedules B››
  apply (case_tac "a ∈ int A")
  apply (drule intA_is_not_actB)
  apply (assumption) (* ⟶ a ∉ act B *)
  apply simp

  text ‹case ‹a ∉ int A›, i.e. ‹a ∈ out A››
  apply (case_tac "a ∉ act B")
  apply simp
  text ‹case ‹a ∈ act B››
  apply simp
  apply (subgoal_tac "a ∈ out A")
  prefer 2
  apply blast
  apply (drule outAactB_is_inpB)
  apply assumption
  apply assumption
  apply (rule scheds_input_enabled)
  apply simp
  apply assumption+
  done

end