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Bridge.v
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Require Import Bool Arith List CpdtTactics SfLib LibTactics Omega.
Require Import Coq.Program.Equality.
Set Implicit Arguments.
Require Import Identifier Environment Imperative Types Augmented.
Inductive low_event : typenv -> level -> event -> Prop :=
| low_assigment_is_low_event:
forall Γ ℓ ℓ' x u,
( ℓ' ⊑ ℓ) ->
low_event Γ ℓ (AssignmentEvent ℓ' x u).
Hint Constructors low_event.
Definition high_event Γ ℓ evt := ~low_event Γ ℓ evt.
Definition low_event_step Γ ℓ evt cfg cfg' :=
event_step Γ evt cfg cfg' /\ low_event Γ ℓ evt.
Definition high_event_step Γ ℓ evt cfg cfg' :=
event_step Γ evt cfg cfg' /\ high_event Γ ℓ evt.
Hint Unfold high_event.
Hint Unfold high_event_step.
Hint Unfold low_event_step.
Definition event_low_eq Γ ev1 ev2 :=
(low_event Γ Low ev1 <-> low_event Γ Low ev2)
/\ (low_event Γ Low ev1 -> ev1 = ev2).
Hint Unfold event_low_eq.
Inductive bridge_step_num:
typenv -> level -> config -> config -> event -> nat -> Prop :=
| bridge_low_num:
forall Γ ℓ evt cfg cfg',
low_event_step Γ ℓ evt cfg cfg' ->
bridge_step_num Γ ℓ cfg cfg' evt 0
| bridge_stop_num:
forall Γ ℓ evt cfg cfg',
high_event_step Γ ℓ evt cfg cfg' ->
is_stop cfg' ->
bridge_step_num Γ ℓ cfg cfg' evt 0
| bridge_trans_num:
forall Γ ℓ evt' evt'' cfg cfg' cfg'' n,
high_event_step Γ ℓ evt' cfg cfg' ->
is_not_stop cfg' ->
bridge_step_num Γ ℓ cfg' cfg'' evt'' n ->
bridge_step_num Γ ℓ cfg cfg'' evt'' (S n).
Tactic Notation "bridge_num_cases" tactic(first) ident(c) :=
first;
[ Case_aux c "bridge_low_num" | Case_aux c "bridge_stop_num" | Case_aux c "bridge_trans_num" ].
Notation " t '⇨+/(SL,' Γ ',' obs ',' n ')' t' " :=
(bridge_step_num Γ Low t t' obs n) (at level 40).
Notation " t '↷' '(' Γ ',' obs ',' n ')' t' " :=
(bridge_step_num Γ Low t t' obs n) (at level 40).
(* Multi-step reduction *)
Inductive multi {X:Type} (R: relation X): relation X :=
| multi_refl : forall (x : X), multi R x x
| multi_step : forall (x y z : X),
R x y ->
multi R y z ->
multi R x z.
Hint Resolve multi_refl.
Tactic Notation "multi_cases" tactic(first) ident(c) :=
first;
[ Case_aux c "multi_refl" | Case_aux c "multi_step" ].
Theorem multi_R : forall (X:Type) (R:relation X) (x y : X),
R x y -> (multi R) x y.
Proof.
intros X R x y H.
apply multi_step with y. apply H. apply multi_refl. Qed.
Theorem multi_trans :
forall (X:Type) (R: relation X) (x y z : X),
multi R x y ->
multi R y z ->
multi R x z.
Proof.
intros.
multi_cases (induction H) Case.
Case "multi_refl".
assumption.
Case "multi_step".
apply multi_step with y; try apply IHmulti; assumption.
Qed.
(* Multi-step transitions *)
Definition multistep := multi step.
Notation " t '⇒*' t' " := (multistep t t') (at level 40).
(* Indexed multi relation *)
Definition relation_idx := fun X : Type => X -> X -> nat -> Prop.
Inductive multi_idx {X:Type} (R: relation X) :relation_idx X :=
| multi_refl_zero : forall (x : X), multi_idx R x x 0
| multi_step_more : forall (x y z : X) n,
R x y ->
multi_idx R y z n ->
multi_idx R x z (S n).
Hint Resolve multi_refl_zero.
Tactic Notation "multi_idx_cases" tactic(first) ident(c) :=
first;
[ Case_aux c "multi_refl_zero" | Case_aux c "multi_step_more" ].
(* Indexed multi-step transitions *)
Definition multistep_idx := multi_idx step.
Notation " t '⇒/+' n '+/' s" := (multi_idx step t s n) (at level 40).
(* A pair of sanity check theorems; we don't really use them *)
Theorem multi_idx_R:
forall (X:Type) (R: relation X) ( x y : X),
R x y -> (multi_idx R) x y 1.
Proof.
intros.
eapply multi_step_more; eauto.
Qed.
Theorem multi_idx_trans:
forall n (X: Type) (R: relation X) ( x y z: X ) m,
multi_idx R x y n ->
multi_idx R y z m ->
multi_idx R x z (n + m).
Proof.
intros n X R.
induction n; intros.
- inverts* H.
- inversion H; subst; auto.
assert (multi_idx R x y0 1) by (econstructor; eauto).
specialize (IHn y0 y z m H5 H0).
replace (S n + m) with (S (n + m)) by omega.
eapply multi_step_more; eauto.
Qed.
(* A relation from multi to indexed multi; this is used in the proof of Standard TINI *)
Theorem from_multi_to_multi_idx:
forall (X:Type) (R:relation X) (x y : X),
multi R x y -> exists n, (multi_idx R x y n).
Proof.
intros.
induction H.
- (exists 0; constructor).
- destruct IHmulti as [n ?IHmulti].
(exists (S n)).
econstructor; eauto.
Qed.