Library OracleLanguages.Imp.Oracles.CrashFix
Require Import Common.
Require Import Imp.Lang.
Require Import ProgrammingLanguage.
Require Import OracleLanguage.
Require Import OracleHelpers.
Module Make
(Var_as_DT : UsualDecidableTypeOrig)
(Import Lang : ImpProgrammingLanguage Var_as_DT)
(Import OracleHelpers : ImpOracleHelpers Var_as_DT Lang).
Record mod := {
condition :
bexp;
c_then :
cmd;
c_else :
cmd;
crash_bound:
store → nat;
crash_avoided_1:
∀ s, eval_bexp s condition ≠ Some true →
∀ k',
nsteps υ (crash_bound s) (c_then::k', s) = None;
}.
Inductive _cont_mod : Type := .
Definition _genv := unit.
Inductive _state : Type :=
| StuckLeft : state → state → _state
| State : state → state → list (cont_mod mod _cont_mod) → _state.
Definition _step (_ : _genv) (S : _state) : option _state :=
match S with
| StuckLeft S₁ S₂ ⇒
step υ S₂ >>= λ S₂',
Some (StuckLeft S₁ S₂')
| State S₁ S₂ km ⇒
match step_helper S₁ S₂ km with
| None ⇒
None
| Some StuckStep ⇒
None
| Some (GenericStep S₁' S₂' km') ⇒
Some (State S₁' S₂' km')
| Some (SpecialCmdStep m km') ⇒
match eval_bexp (snd S₂) (condition m) with
| None ⇒
let b := crash_bound m (snd S₁) in
let (S₁', n') := nsteps' υ b S₁ in
match (b - n') with
| O ⇒
None
| _ ⇒
Some (StuckLeft S₁' S₂)
end
| Some true ⇒
step υ S₂ >>= λ S₂',
Some (State S₁ S₂' (CmdMod Id::km'))
| Some false ⇒
step υ S₂ >>= λ S₂',
let b := crash_bound m (snd S₁) in
Some (StuckLeft (fst (nsteps' υ b S₁)) S₂')
end
| Some (SpecialContStep _ km') ⇒
None
end
end.
Definition _left_state (S : _state) : ProgrammingLanguage.state imp_language :=
match S with
| StuckLeft S₁ _ | State S₁ _ _ ⇒
S₁
end.
Definition _right_state (S : _state) : ProgrammingLanguage.state imp_language :=
match S with
| StuckLeft _ S₂ | State _ S₂ _ ⇒
S₂
end.
Inductive mod_invariant : cmd_mod mod → cmd → cmd → Prop :=
| mod_id :
∀ c,
mod_invariant Id c c
| mod_rec_seq :
∀ m₁ m₂ c₁₁ c₁₂ c₂₁ c₂₂,
mod_invariant m₁ c₁₁ c₂₁ →
mod_invariant m₂ c₁₂ c₂₂ →
mod_invariant (RecMod m₁ m₂) (Seq c₁₁ c₁₂) (Seq c₂₁ c₂₂)
| mod_rec_ite :
∀ b m₁ m₂ c₁₁ c₁₂ c₂₁ c₂₂,
mod_invariant m₁ c₁₁ c₂₁ →
mod_invariant m₂ c₁₂ c₂₂ →
mod_invariant (RecMod m₁ m₂) (ITE b c₁₁ c₁₂) (ITE b c₂₁ c₂₂)
| mod_rec_while :
∀ b m m' c₁ c₂,
mod_invariant m c₁ c₂ →
mod_invariant (RecMod m m') (While b c₁) (While b c₂)
| mod_special :
∀ m,
mod_invariant (LeafMod m) (c_then m) (ITE (condition m) (c_then m) (c_else m)).
Inductive km_invariant : list (cont_mod mod _cont_mod) → cont → cont → Prop :=
| empty_cont_mod : km_invariant [] [] []
| cmd_cont : ∀ c c' m km k₁ k₂,
mod_invariant m c c' →
km_invariant km k₁ k₂ →
km_invariant (CmdMod m::km) (c::k₁) (c'::k₂).
Definition _invariant (_ : _genv) (S : _state) : Prop :=
match S with
| StuckLeft S₁ S₂ ⇒ step υ S₁ = None
| State (k₁, s₁) (k₂, s₂) km ⇒
km_invariant km k₁ k₂ ∧ M.Equal s₁ s₂
end.
Lemma _invariant_1:
∀ genv os os',
_invariant genv os →
_step genv os = Some os' →
_invariant genv os'.
Lemma _prediction_soundness :
∀ genv os os',
_invariant genv os →
_step genv os = Some os' →
∃ n₁ n₂,
opt_state_eq (nsteps υ n₁ (_left_state os))
(Some (_left_state os'))
∧ opt_state_eq (nsteps υ n₂ (_right_state os))
(Some (_right_state os'))
∧ n₁ + n₂ > 0.
Lemma _prediction_completeness :
∀ genv os,
_invariant genv os →
_step genv os = None →
step υ (_left_state os) = None
∧ step υ (_right_state os) = None.
Definition crashfix_oracle : oracle_language imp_language imp_language :=
{|
oracle_state := _state;
oracle_genv := unit;
oracle_step := _step;
left_state := _left_state;
right_state := _right_state;
left_genv := λ _, υ;
right_genv := λ _, υ;
invariant := _invariant;
invariant_1 := _invariant_1;
prediction_soundness := _prediction_soundness;
prediction_completeness := _prediction_completeness
|}.
End Make.