Theory Traces

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

section ‹Executions and Traces of I/O automata in HOLCF›

theory Traces
imports Sequence Automata
begin

default_sort type

type_synonym ('a, 's) pairs = "('a × 's) Seq"
type_synonym ('a, 's) execution = "'s × ('a, 's) pairs"
type_synonym 'a trace = "'a Seq"
type_synonym ('a, 's) execution_module = "('a, 's) execution set × 'a signature"
type_synonym 'a schedule_module = "'a trace set × 'a signature"
type_synonym 'a trace_module = "'a trace set × 'a signature"


subsection ‹Executions›

definition is_exec_fragC :: "('a, 's) ioa ⇒ ('a, 's) pairs → 's ⇒ tr"
  where "is_exec_fragC A =
    (fix ⋅
      (LAM h ex.
        (λs.
          case ex of
            nil ⇒ TT
          | x ## xs ⇒ flift1 (λp. Def ((s, p) ∈ trans_of A) andalso (h ⋅ xs) (snd p)) ⋅ x)))"

definition is_exec_frag :: "('a, 's) ioa ⇒ ('a, 's) execution ⇒ bool"
  where "is_exec_frag A ex ⟷ (is_exec_fragC A ⋅ (snd ex)) (fst ex) ≠ FF"

definition executions :: "('a, 's) ioa ⇒ ('a, 's) execution set"
  where "executions ioa = {e. fst e ∈ starts_of ioa ∧ is_exec_frag ioa e}"


subsection ‹Schedules›

definition filter_act :: "('a, 's) pairs → 'a trace"
  where "filter_act = Map fst"

definition has_schedule :: "('a, 's) ioa ⇒ 'a trace ⇒ bool"
  where "has_schedule ioa sch ⟷ (∃ex ∈ executions ioa. sch = filter_act ⋅ (snd ex))"

definition schedules :: "('a, 's) ioa ⇒ 'a trace set"
  where "schedules ioa = {sch. has_schedule ioa sch}"


subsection ‹Traces›

definition has_trace :: "('a, 's) ioa ⇒ 'a trace ⇒ bool"
  where "has_trace ioa tr ⟷ (∃sch ∈ schedules ioa. tr = Filter (λa. a ∈ ext ioa) ⋅ sch)"

definition traces :: "('a, 's) ioa ⇒ 'a trace set"
  where "traces ioa ≡ {tr. has_trace ioa tr}"

definition mk_trace :: "('a, 's) ioa ⇒ ('a, 's) pairs → 'a trace"
  where "mk_trace ioa = (LAM tr. Filter (λa. a ∈ ext ioa) ⋅ (filter_act ⋅ tr))"


subsection ‹Fair Traces›

definition laststate :: "('a, 's) execution ⇒ 's"
  where "laststate ex =
    (case Last ⋅ (snd ex) of
      UU ⇒ fst ex
    | Def at ⇒ snd at)"

text ‹A predicate holds infinitely (finitely) often in a sequence.›
definition inf_often :: "('a ⇒ bool) ⇒ 'a Seq ⇒ bool"
  where "inf_often P s ⟷ Infinite (Filter P ⋅ s)"

text ‹Filtering ‹P› yields a finite or partial sequence.›
definition fin_often :: "('a ⇒ bool) ⇒ 'a Seq ⇒ bool"
  where "fin_often P s ⟷ ¬ inf_often P s"


subsection ‹Fairness of executions›

text ‹
  Note that partial execs cannot be ‹wfair› as the inf_often predicate in the
  else branch prohibits it. However they can be ‹sfair› in the case when all
  ‹W› are only finitely often enabled: Is this the right model?

  See 🗏‹LiveIOA.thy› for solution conforming with the literature and
  superseding this one.
›

definition is_wfair :: "('a, 's) ioa ⇒ 'a set ⇒ ('a, 's) execution ⇒ bool"
  where "is_wfair A W ex ⟷
    (inf_often (λx. fst x ∈ W) (snd ex) ∨
      inf_often (λx. ¬ Enabled A W (snd x)) (snd ex))"

definition wfair_ex :: "('a, 's) ioa ⇒ ('a, 's) execution ⇒ bool"
  where "wfair_ex A ex ⟷
    (∀W ∈ wfair_of A.
      if Finite (snd ex)
      then ¬ Enabled A W (laststate ex)
      else is_wfair A W ex)"

definition is_sfair :: "('a, 's) ioa ⇒ 'a set ⇒ ('a, 's) execution ⇒ bool"
  where "is_sfair A W ex ⟷
    (inf_often (λx. fst x ∈ W) (snd ex) ∨
      fin_often (λx. Enabled A W (snd x)) (snd ex))"

definition sfair_ex :: "('a, 's)ioa ⇒ ('a, 's) execution ⇒ bool"
  where "sfair_ex A ex ⟷
    (∀W ∈ sfair_of A.
      if Finite (snd ex)
      then ¬ Enabled A W (laststate ex)
      else is_sfair A W ex)"

definition fair_ex :: "('a, 's) ioa ⇒ ('a, 's) execution ⇒ bool"
  where "fair_ex A ex ⟷ wfair_ex A ex ∧ sfair_ex A ex"


text ‹Fair behavior sets.›

definition fairexecutions :: "('a, 's) ioa ⇒ ('a, 's) execution set"
  where "fairexecutions A = {ex. ex ∈ executions A ∧ fair_ex A ex}"

definition fairtraces :: "('a, 's) ioa ⇒ 'a trace set"
  where "fairtraces A = {mk_trace A ⋅ (snd ex) | ex. ex ∈ fairexecutions A}"


subsection ‹Implementation›

subsubsection ‹Notions of implementation›

definition ioa_implements :: "('a, 's1) ioa ⇒ ('a, 's2) ioa ⇒ bool"  (infixr "=<|" 12)
  where "(ioa1 =<| ioa2) ⟷
    (inputs (asig_of ioa1) = inputs (asig_of ioa2) ∧
     outputs (asig_of ioa1) = outputs (asig_of ioa2)) ∧
    traces ioa1 ⊆ traces ioa2"

definition fair_implements :: "('a, 's1) ioa ⇒ ('a, 's2) ioa ⇒ bool"
  where "fair_implements C A ⟷
    inp C = inp A ∧ out C = out A ∧ fairtraces C ⊆ fairtraces A"

lemma implements_trans: "A =<| B ⟹ B =<| C ⟹ A =<| C"
  by (auto simp add: ioa_implements_def)


subsection ‹Modules›

subsubsection ‹Execution, schedule and trace modules›

definition Execs :: "('a, 's) ioa ⇒ ('a, 's) execution_module"
  where "Execs A = (executions A, asig_of A)"

definition Scheds :: "('a, 's) ioa ⇒ 'a schedule_module"
  where "Scheds A = (schedules A, asig_of A)"

definition Traces :: "('a, 's) ioa ⇒ 'a trace_module"
  where "Traces A = (traces A, asig_of A)"

lemmas [simp del] = HOL.ex_simps HOL.all_simps split_paired_Ex
declare Let_def [simp]
setup ‹map_theory_claset (fn ctxt => ctxt delSWrapper "split_all_tac")›

lemmas exec_rws = executions_def is_exec_frag_def


subsection ‹Recursive equations of operators›

subsubsection ‹‹filter_act››

lemma filter_act_UU: "filter_act ⋅ UU = UU"
  by (simp add: filter_act_def)

lemma filter_act_nil: "filter_act ⋅ nil = nil"
  by (simp add: filter_act_def)

lemma filter_act_cons: "filter_act ⋅ (x ↝ xs) = fst x ↝ filter_act ⋅ xs"
  by (simp add: filter_act_def)

declare filter_act_UU [simp] filter_act_nil [simp] filter_act_cons [simp]


subsubsection ‹‹mk_trace››

lemma mk_trace_UU: "mk_trace A ⋅ UU = UU"
  by (simp add: mk_trace_def)

lemma mk_trace_nil: "mk_trace A ⋅ nil = nil"
  by (simp add: mk_trace_def)

lemma mk_trace_cons:
  "mk_trace A ⋅ (at ↝ xs) =
    (if fst at ∈ ext A
     then fst at ↝ mk_trace A ⋅ xs
     else mk_trace A ⋅ xs)"
  by (simp add: mk_trace_def)

declare mk_trace_UU [simp] mk_trace_nil [simp] mk_trace_cons [simp]


subsubsection ‹‹is_exec_fragC››

lemma is_exec_fragC_unfold:
  "is_exec_fragC A =
    (LAM ex.
      (λs.
        case ex of
          nil ⇒ TT
        | x ## xs ⇒
            (flift1 (λp. Def ((s, p) ∈ trans_of A) andalso (is_exec_fragC A⋅xs) (snd p)) ⋅ x)))"
  apply (rule trans)
  apply (rule fix_eq4)
  apply (rule is_exec_fragC_def)
  apply (rule beta_cfun)
  apply (simp add: flift1_def)
  done

lemma is_exec_fragC_UU: "(is_exec_fragC A ⋅ UU) s = UU"
  apply (subst is_exec_fragC_unfold)
  apply simp
  done

lemma is_exec_fragC_nil: "(is_exec_fragC A ⋅ nil) s = TT"
  apply (subst is_exec_fragC_unfold)
  apply simp
  done

lemma is_exec_fragC_cons:
  "(is_exec_fragC A ⋅ (pr ↝ xs)) s =
    (Def ((s, pr) ∈ trans_of A) andalso (is_exec_fragC A ⋅ xs) (snd pr))"
  apply (rule trans)
  apply (subst is_exec_fragC_unfold)
  apply (simp add: Consq_def flift1_def)
  apply simp
  done

declare is_exec_fragC_UU [simp] is_exec_fragC_nil [simp] is_exec_fragC_cons [simp]


subsubsection ‹‹is_exec_frag››

lemma is_exec_frag_UU: "is_exec_frag A (s, UU)"
  by (simp add: is_exec_frag_def)

lemma is_exec_frag_nil: "is_exec_frag A (s, nil)"
  by (simp add: is_exec_frag_def)

lemma is_exec_frag_cons:
  "is_exec_frag A (s, (a, t) ↝ ex) ⟷ (s, a, t) ∈ trans_of A ∧ is_exec_frag A (t, ex)"
  by (simp add: is_exec_frag_def)

declare is_exec_frag_UU [simp] is_exec_frag_nil [simp] is_exec_frag_cons [simp]


subsubsection ‹‹laststate››

lemma laststate_UU: "laststate (s, UU) = s"
  by (simp add: laststate_def)

lemma laststate_nil: "laststate (s, nil) = s"
  by (simp add: laststate_def)

lemma laststate_cons: "Finite ex ⟹ laststate (s, at ↝ ex) = laststate (snd at, ex)"
  apply (simp add: laststate_def)
  apply (cases "ex = nil")
  apply simp
  apply simp
  apply (drule Finite_Last1 [THEN mp])
  apply assumption
  apply defined
  done

declare laststate_UU [simp] laststate_nil [simp] laststate_cons [simp]

lemma exists_laststate: "Finite ex ⟹ ∀s. ∃u. laststate (s, ex) = u"
  by Seq_Finite_induct


subsection ‹‹has_trace› ‹mk_trace››

(*alternative definition of has_trace tailored for the refinement proof, as it does not
  take the detour of schedules*)
lemma has_trace_def2: "has_trace A b ⟷ (∃ex ∈ executions A. b = mk_trace A ⋅ (snd ex))"
  apply (unfold executions_def mk_trace_def has_trace_def schedules_def has_schedule_def [abs_def])
  apply auto
  done


subsection ‹Signatures and executions, schedules›

text ‹
  All executions of ‹A› have only actions of ‹A›. This is only true because of
  the predicate ‹state_trans› (part of the predicate ‹IOA›): We have no
  dependent types. For executions of parallel automata this assumption is not
  needed, as in ‹par_def› this condition is included once more. (See Lemmas
  1.1.1c in CompoExecs for example.)
›

lemma execfrag_in_sig:
  "is_trans_of A ⟹ ∀s. is_exec_frag A (s, xs) ⟶ Forall (λa. a ∈ act A) (filter_act ⋅ xs)"
  apply (pair_induct xs simp: is_exec_frag_def Forall_def sforall_def)
  text ‹main case›
  apply (auto simp add: is_trans_of_def)
  done

lemma exec_in_sig:
  "is_trans_of A ⟹ x ∈ executions A ⟹ Forall (λa. a ∈ act A) (filter_act ⋅ (snd x))"
  apply (simp add: executions_def)
  apply (pair x)
  apply (rule execfrag_in_sig [THEN spec, THEN mp])
  apply auto
  done

lemma scheds_in_sig: "is_trans_of A ⟹ x ∈ schedules A ⟹ Forall (λa. a ∈ act A) x"
  apply (unfold schedules_def has_schedule_def [abs_def])
  apply (fast intro!: exec_in_sig)
  done


subsection ‹Executions are prefix closed›

(*only admissible in y, not if done in x!*)
lemma execfrag_prefixclosed: "∀x s. is_exec_frag A (s, x) ∧ y ⊑ x ⟶ is_exec_frag A (s, y)"
  apply (pair_induct y simp: is_exec_frag_def)
  apply (intro strip)
  apply (Seq_case_simp x)
  apply (pair a)
  apply auto
  done

lemmas exec_prefixclosed =
  conjI [THEN execfrag_prefixclosed [THEN spec, THEN spec, THEN mp]]

(*second prefix notion for Finite x*)
lemma exec_prefix2closed [rule_format]:
  "∀y s. is_exec_frag A (s, x @@ y) ⟶ is_exec_frag A (s, x)"
  apply (pair_induct x simp: is_exec_frag_def)
  apply (intro strip)
  apply (Seq_case_simp s)
  apply (pair a)
  apply auto
  done

end