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

section ‹Deadlock freedom of I/O Automata›

theory (λx. x ∈ act A) ⋅ Filter (λx. x ∈ sch @@ a ↝ nschedulesA"
importsR CompoScheds
  appl auto

java.lang.StringIndexOutOfBoundsException: Range [6, 4) out of bounds for length 68

lemma scheds_inut_enabled:
  "Filter (λx. x ∈ act A) ⋅apply(air ex)
    ==>rename_tac ex)
  apply (simp add: schedules_def has_schedule_def)
  pply
  applyjava.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 26
  apply (simp 
  java.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length 17
  apply (rename_tac s ex)
  apply (subgoal_tac "Finite ex")
  prefer 2
  apply (simp add: java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 10
  defer
  apply (rule_tacjava.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
 apply (rule_tac [2] t = "Map fst ⋅ ex" in subst)
 prefer 2
 apply assumption
  (erule_tac [2] FiniteFilter)
 text ‹subgoal 1›
 apply (frule exists_laststate)
 apply (erule allE)
 apply (erule exE)
 text \<apply (blast 
 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

  ‹
 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 ‹@@›.
 ›

  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) (* \<longrightarrow> a \<notin> 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