From Undecidability Require Import Hoare.HoareLogic.
From Undecidability.TM Require Import TMTac.
From Undecidability.TM Require Export CodeTM LiftTapes ChangeAlphabet.
Section RegSpec.
Variable sig : Type.
Local Notation "sig '^+'" := ((boundary + sig) % type) (at level 0) : type_scope.
Inductive RegSpec : Type :=
| Contains {sigX X : Type} {cX : codable sigX X} (r : Retract sigX sig) : X -> RegSpec
| Contains_size {sigX X : Type} {cX : codable sigX X} (r : Retract sigX sig) : X -> nat -> RegSpec
| Void : RegSpec
| Void_size : nat -> RegSpec
| Custom : (tape sig^+ -> Prop) -> RegSpec.
Definition tspec_single (spec : RegSpec) (t : tape sig^+) : Prop :=
match spec with
| Contains r x => t ≃(r) x
| Contains_size r x s => t ≃(r; s) x
| Void => isVoid t
| Void_size s => isVoid_size t s
| Custom p => p t
end.
Lemma tspec_single_Contains_size_Contains {sigX X : Type} {cX : codable sigX X} (r : Retract sigX sig) (x : X) (s : nat) (t : tape sig^+) :
tspec_single (Contains_size r x s) t -> tspec_single (Contains r x) t.
Proof. cbn. auto. Qed.
Definition SpecV n := Vector.t (RegSpec) n.
Definition Spec n :Type := list Prop * SpecV n.
Definition tspec {n : nat} (spec : Spec n) : Assert sig^+ n :=
match spec with
| (P,spec) => fun (t : tapes sig^+ n) =>
List.fold_right and (forall (i : Fin.t n), tspec_single spec[@i] t[@i]) P
end.
Arguments tspec : simpl never.
Lemma tspecE {n : nat} Ps Pv t:
tspec (n:=n) (Ps,Pv) t -> (List.fold_right and True Ps /\ forall (i : Fin.t n), tspec_single Pv[@i] t[@i]).
Proof.
cbn. induction Ps;cbn in *. easy. intros []. specialize (IHPs H0) as [H' ?]. split. all:eauto.
Qed.
Lemma tspecI {n : nat} P t:
(List.fold_right and True (fst P))
-> (forall (i : Fin.t n), tspec_single (snd P)[@i] t[@i])
-> tspec (n:=n) P t.
Proof.
destruct P as (Ps&P). induction Ps;cbn in *. easy. intros H' ?. split. now eapply H';left.
eapply IHPs. all:easy.
Qed.
Lemma tspec_iff {n : nat} Ps Pv t:
tspec (n:=n) (Ps,Pv) t <-> (List.fold_right and True Ps /\ forall (i : Fin.t n), tspec_single Pv[@i] t[@i]).
Proof. split. eapply tspecE;eauto. intros []. eapply tspecI;eauto. Qed.
Lemma tspec_Entails {n : nat} Ps Pv:
Entails (tspec (n:=n) (Ps,Pv)) (fun t => List.fold_right and True Ps /\ forall (i : Fin.t n), tspec_single Pv[@i] t[@i]).
Proof. apply EntailsI. now setoid_rewrite tspec_iff. Qed.
Definition withSpace_single (P : RegSpec) (size : nat) :=
match P with
| Contains r x => Contains_size r x size
| Void => Void_size size
| _ => P
end.
Definition withSpace {n : nat} (P : SpecV n) (spaces : Vector.t nat n) : SpecV n :=
Vector.map2 withSpace_single P spaces.
Lemma tspec_single_withSpace_tspec_single (P : RegSpec) (size : nat) t :
tspec_single (withSpace_single P size) t -> tspec_single P t.
Proof. intros. destruct P; cbn in *; auto. Qed.
Lemma tspec_withSpace_tspec {n : nat} Q P (s : Vector.t nat n) t :
tspec (Q,withSpace P s) t -> tspec (Q,P) t.
Proof. unfold withSpace. intros [HP H]%tspecE. apply tspecI. easy. intros i; specialize (H i). simpl_vector in *. eapply tspec_single_withSpace_tspec_single; eauto. Qed.
Definition dummy_size (t : tape sig^+) (P : RegSpec) : nat :=
match P with
| Contains r x => length (left t)
| Void => length (tape_local_l t)
| _ => 0
end.
Lemma tspec_single_tspec_single_withSpace (P : RegSpec) t :
tspec_single P t -> tspec_single (withSpace_single P (dummy_size t P)) t.
Proof. intros H. destruct P; cbn in *; eauto. Qed.
Definition dummy_sizes (n : nat) (t : tapes sig^+ n) (P : SpecV n) : Vector.t nat n :=
Vector.map2 dummy_size t P.
Lemma tspec_tspec_withSpace (n : nat) Q (P : SpecV n) t :
tspec (Q,P) t -> tspec (Q,withSpace P (dummy_sizes t P)) t.
Proof.
unfold withSpace, dummy_sizes in *. intros [HP ?]%tspecE. eapply tspecI. cbn in *; auto.
intros i; specialize (H i); cbn in *. simpl_vector. now apply tspec_single_tspec_single_withSpace.
Qed.
End RegSpec.
Arguments Custom {sig}.
Arguments Void {sig}.
Arguments Void_size {sig}.
Arguments dummy_sizes : simpl never.
Hint Resolve tspec_single_Contains_size_Contains : core.
Declare Scope spec_scope.
Delimit Scope spec_scope with spec.
Bind Scope spec_scope with Spec.
Notation "'≃≃(' S ')'" := (tspec S%spec) (at level 0, S at level 200, no associativity, format "'≃≃(' S ')'").
Notation "'≃≃(' P ',' S ')'" := (tspec (P%list,S%vector)) (at level 0, P at level 200, S at level 200, no associativity, format "'≃≃(' P ',' S ')'").
Notation "t ≃≃ S" := (tspec S%spec t) (at level 70, no associativity).
Notation "≃( I ) x" := (Contains I x) (at level 70, I at level 200, no associativity, format "'≃(' I ')' x").
Notation "≃( I ';' s ) x" := (Contains_size I s x) (at level 70, I at level 200, s at level 200, no associativity, format "'≃(' I ';' s ')' x").
Arguments tspec _%spec _.
Fixpoint implList (Ps : list Prop) (Q : Prop) :=
match Ps with
[] => Q
| P::Ps => P -> implList Ps Q
end.
Arguments implList !_ _.
Instance fold_right_impl' : Proper (Forall2 Basics.impl --> Basics.impl ==> Basics.impl) (implList).
Proof. intros xs;induction xs;cbn;intros ? H';inv H';cbn. easy. firstorder. Qed.
Instance fold_right_iff : Proper (Forall2 iff ==> iff ==> iff) (implList).
Proof. intros xs;induction xs;cbn;intros ? H';inv H';cbn. easy. firstorder. Qed.
Lemma implList_iff P (Ps : list Prop):
implList Ps P
<-> (List.fold_right and True Ps -> P).
Proof.
induction Ps in P|-*;cbn. firstorder. setoid_rewrite IHPs. tauto.
Qed.
Lemma implListE P (Ps : list Prop): implList Ps P -> (List.fold_right and True Ps -> P).
Proof. now rewrite implList_iff. Qed.
Lemma implListI (P:Prop) (Ps : list Prop): (List.fold_right and True Ps -> P) -> implList Ps P.
Proof. now rewrite implList_iff. Qed.
Instance Forall2_refl X (R: X -> _): Reflexive R -> Reflexive (Forall2 R).
Proof. intros ? xs;induction xs;eauto. Qed.
Lemma tspec_introPure (sig: finType) (n : nat) P (Ps : SpecV sig n) Q:
implList P (Entails (≃≃([],Ps)) Q)
-> Entails (tspec (P,Ps)) Q.
Proof.
setoid_rewrite Entails_iff. rewrite implList_iff. intros H ? []%tspecE. eapply H. eassumption. now apply tspecI.
Qed.
Lemma tspec_revertPure (sig: finType) (n : nat) (P0:Prop) P (Ps : SpecV sig n) Q:
P0
-> Entails (tspec (P0::P,Ps)) Q
-> Entails (tspec (P,Ps)) Q.
Proof.
setoid_rewrite Entails_iff. unfold tspec;cbn. intuition.
Qed.
Lemma Triple_introPure (F sig: finType) (n : nat) P (Ps : SpecV sig n) Q (pM : pTM sig^+ F n) :
implList P (Triple (≃≃([],Ps)) pM Q)
-> Triple (tspec (P,Ps)) pM Q.
Proof.
intros. rewrite tspec_Entails. apply Triple_and_pre. cbn in H. now rewrite <- implList_iff.
Qed.
Lemma TripleT_introPure (sig F : finType) (n : nat) P (Ps : SpecV sig n) Q k (pM : pTM sig^+ F n) :
implList P (TripleT (≃≃([],Ps)) k pM Q)
-> TripleT (tspec (P,Ps)) k pM Q.
Proof.
intros. rewrite tspec_Entails. apply TripleT_and_pre. cbn in H. now rewrite <- implList_iff.
Qed.
Definition SpecVTrue {sig : Type} {n : nat} : SpecV sig n := Vector.const (Custom (fun _ => True)) n.
Arguments tspec : simpl never.
Definition appSize {n : nat} : Vector.t (nat->nat) n -> Vector.t nat n -> Vector.t nat n :=
fun fs s => tabulate (fun i => fs[@i] s[@i]).
Lemma Triple_RemoveSpace_ex (n : nat) (sig : finType) (F : Type) X
(P : SpecV sig n) P' (M : pTM sig^+ F n) Q Q' Ctx (fs : _ -> Vector.t (nat->nat) n) :
(forall s, Triple (tspec (P',withSpace P s)) M (fun y t => exists x:X, Ctx x (tspec (Q' x y,withSpace (Q x y) (appSize (fs x) s)) t))) ->
(forall x, Proper (Basics.impl ==> Basics.impl) (Ctx x)) ->
Triple (tspec (P',P)) M (fun y t => exists x, Ctx x (tspec (Q' x y ,Q x y) t)).
Proof.
intros HTrip Hctx. setoid_rewrite Triple_iff in HTrip. rewrite Triple_iff.
eapply Realise_monotone with
(R' := fun tin '(yout, tout) => forall s, tspec (P',withSpace P s) tin
-> exists x:X, Ctx x (tspec (Q' x yout,withSpace (Q x yout) (appSize (fs x) s)) tout)).
- unfold Triple_Rel, Realise in *. intros tin k outc HLoop. intros s HP.
specialize HTrip with (1 := HLoop) (2 := HP) as [x H'']. eexists. eauto.
- clear HTrip. intros tin (yout, tout). intros H HP.
specialize (H (dummy_sizes tin P)). spec_assert H by now apply tspec_tspec_withSpace.
destruct H as (x&H). exists x. eapply Hctx. 2:eassumption.
intro H'. now apply tspec_withSpace_tspec in H'.
Qed.
Lemma Triple_RemoveSpace (n : nat) (sig : finType) (F : Type) (P : SpecV sig n) P' (M : pTM sig^+ F n) (Q : F -> SpecV sig n) Q' (fs : Vector.t (nat->nat) n) :
(forall s, Triple (tspec (P',withSpace P s)) M (fun y => tspec (Q' y,withSpace (Q y) (appSize fs s)))) ->
Triple (tspec (P',P)) M (fun y => tspec (Q' y ,Q y)).
Proof.
intro HTrip. setoid_rewrite Triple_iff in HTrip. rewrite Triple_iff.
eapply Realise_monotone with (R' := fun tin '(yout, tout) => forall s, tspec (P',withSpace P s) tin -> tspec (Q' yout,withSpace (Q yout) (appSize fs s)) tout).
- unfold Triple_Rel, Realise in *. intros tin k outc HLoop. intros s HP.
now specialize HTrip with (1 := HLoop) (2 := HP).
- clear HTrip. intros tin (yout, tout). intros H HP.
specialize (H (dummy_sizes tin P)). spec_assert H by now apply tspec_tspec_withSpace.
now apply tspec_withSpace_tspec in H.
Qed.
Lemma TripleT_RemoveSpace (n : nat) (sig : finType) (F : Type) P' (P : SpecV sig n) (k : nat) (M : pTM sig^+ F n) Q' (Q : F -> SpecV sig n) (fs : Vector.t (nat->nat) n) :
(forall s, TripleT (tspec (P',withSpace P s)) k M (fun y => tspec (Q' y,withSpace (Q y) (appSize fs s)))) ->
TripleT (tspec (P',P)) k M (fun y => tspec (Q' y,Q y)).
Proof.
intros HTrip. split.
- eapply Triple_RemoveSpace. intros s. apply HTrip.
- setoid_rewrite TripleT_iff in HTrip.
eapply TerminatesIn_monotone with (T' := fun tin k' => tspec (P',P) tin /\ k <= k').
+ unfold Triple_TRel, TerminatesIn in *. intros tin k' (HP&Hk).
specialize (HTrip (dummy_sizes tin P)) as (_&HT).
specialize HT with (tin0 := tin) (k0 := k). spec_assert HT as (conf&HLoop).
{ split. now apply tspec_tspec_withSpace. reflexivity. }
exists conf. eapply loop_monotone; eauto.
+ unfold Triple_TRel. intros tin k' (HP&Hk). eauto.
Qed.
Instance fold_right_and : Proper (iff ==> Forall2 iff ==> iff) (fold_right and).
Proof. intros ? ? ? xs;induction xs;cbn;intros ? H';inv H';cbn. easy. firstorder. Qed.
Instance fold_right_and' : Proper (Basics.impl ==> Forall2 iff ==> Basics.impl) (fold_right and).
Proof. intros ? ? ? xs;induction xs;cbn;intros ? H';inv H';cbn. easy. firstorder. Qed.
Lemma tspec_solve (sig : Type) (n : nat) (t : tapes (boundary+sig) n) (R : SpecV sig n) P:
List.fold_right and (forall i, tspec_single R[@i] t[@i]) P ->
tspec (P,R) t.
Proof. refine (fun P => P). Qed.
Lemma tspec_space_solve (sig : Type) (n : nat) (t : tapes (boundary+sig) n) (R : SpecV sig n) P (ss : Vector.t nat n) :
List.fold_right and (forall i, tspec_single (withSpace_single R[@i] ss[@i]) t[@i]) P ->
tspec (P,withSpace R ss) t.
Proof. unfold withSpace. intros. apply tspec_solve. simpl_vector. auto. Qed.
Lemma tspec_ext (sig : finType) (n : nat) (t : tapes (boundary+sig) n) (P P' : list Prop) (R R' : Vector.t (RegSpec sig) n) :
tspec (P',R') t ->
implList P' (List.fold_right and True P) ->
(forall i, tspec_single R'[@i] t[@i] -> tspec_single R[@i] t[@i]) ->
tspec (P,R) t.
Proof.
intros [HP H1]%tspecE H1' H2. eapply tspecI.
eapply implList_iff. 2:eassumption. eapply fold_right_impl'. 2:reflexivity. 2:eassumption. easy.
intros i; specialize (H1 i); specialize (H2 i); eauto.
Qed.
Lemma tspec_space_ext (sig : finType) (n : nat) (t : tapes (boundary+sig) n) (P P':list Prop) (R R' : SpecV sig n)
(ss ss' : Vector.t nat n) :
tspec (P',withSpace R' ss') t ->
implList P' (List.fold_right and True P) ->
(forall i, tspec_single (withSpace_single R'[@i] ss'[@i]) t[@i] -> tspec_single (withSpace_single R[@i] ss[@i]) t[@i]) ->
tspec (P,withSpace R ss) t.
Proof.
unfold withSpace. intros [HP H1]%tspecE H1' H2. eapply tspecI.
eapply implList_iff. 2:eassumption. eapply fold_right_impl'. 2:reflexivity. 2:eassumption. easy.
intros i; specialize (H1 i); specialize (H2 i); eauto.
cbn. simpl_vector in *; cbn. eauto.
Qed.
Section Lifting.
Variable (sig : Type).
Variable (m n : nat).
Variable (P : @SpecV sig n).
Variable (I : Vector.t (Fin.t n) m). Hypothesis (HI : dupfree I).
Definition Downlift : @SpecV sig m :=
(select I P).
Lemma tape_fulfill_Downlift_select P' tp :
tspec (P',P) tp ->
tspec (P',Downlift) (select I tp).
Proof.
unfold Downlift.
intros [? H]%tspecE.
eapply tspecI. easy.
intros i;cbn. rewrite !select_nth. easy.
Qed.
Definition Frame (Q : @SpecV sig m) : @SpecV sig n := fill I P Q.
End Lifting.
Lemma LiftTapes_Spec_ex (sig : finType) X (F : finType) (m n : nat) (I : Vector.t (Fin.t n) m)
P' P Q' Q (pM : pTM sig^+ F m) :
dupfree I ->
Triple (tspec (P',Downlift P I)) pM (fun y t => exists x:X, tspec (Q' x y,Q x y) t) ->
Triple (tspec (P',P)) (LiftTapes pM I) (fun y t=> exists x, tspec (Q' x y,Frame P I (Q x y)) t ).
Proof.
unfold Frame. rewrite !Triple_iff.
intros HDup HTrip.
eapply Realise_monotone.
{ apply LiftTapes_Realise. assumption. apply HTrip. }
{
intros tin (yout, tout) (H&HInj). cbn -[Downlift tspec] in *.
intros HP.
spec_assert H by now apply tape_fulfill_Downlift_select.
destruct H as [x H].
eapply tspecE in H as [H' H]. eapply tspecE in HP as [HP' HP].
exists x.
eapply tspecI;cbn.
{ clear - H' HP'. induction P';cbn in *. all:firstorder. }
clear H' HP'.
hnf. intros j. decide (Vector.In j I) as [HD|HD].
- unfold Frame.
apply vect_nth_In' in HD as (ij&HD).
erewrite fill_correct_nth; eauto.
specialize (H ij).
now rewrite select_nth, HD in H.
- unfold Frame. rewrite fill_not_index; eauto.
specialize (HInj j HD). rewrite HInj. now apply HP.
}
Qed.
Lemma LiftTapes_Spec (sig : finType) (F : finType) (m n : nat) (I : Vector.t (Fin.t n) m) P' (P : SpecV sig n) Q' (Q : F -> SpecV sig m) (pM : pTM sig^+ F m) :
dupfree I ->
Triple (tspec (P',Downlift P I)) pM (fun y => tspec (Q' y,Q y)) ->
Triple (tspec (P',P)) (LiftTapes pM I) (fun y => tspec (Q' y,Frame P I (Q y))).
Proof.
unfold Frame. rewrite !Triple_iff.
intros HDup HTrip.
eapply Realise_monotone.
{ apply LiftTapes_Realise. assumption. apply HTrip. }
{
intros tin (yout, tout) (H&HInj). cbn -[Downlift tspec] in *.
intros HP.
spec_assert H by now apply tape_fulfill_Downlift_select.
eapply tspecE in H as [H' H]. eapply tspecE in HP as [HP' HP].
eapply tspecI;cbn.
{ clear - H' HP'. induction P';cbn in *. all:firstorder. }
clear H' HP'.
hnf. intros j. decide (Vector.In j I) as [HD|HD].
- unfold Frame.
apply vect_nth_In' in HD as (ij&HD).
erewrite fill_correct_nth; eauto.
specialize (H ij).
now rewrite select_nth, HD in H.
- unfold Frame. rewrite fill_not_index; eauto.
specialize (HInj j HD). rewrite HInj. now apply HP.
}
Qed.
Lemma LiftTapes_Spec_con (sig : finType) (F : finType) (m n : nat) (I : Vector.t (Fin.t n) m) P' (P : SpecV sig n) Q' (Q : F -> SpecV sig m) R' (R : F -> SpecV sig n) (pM : pTM sig^+ F m) :
dupfree I ->
Triple (tspec (P',Downlift P I)) pM (fun y => tspec (Q' y,Q y)) ->
(forall yout, Entails (tspec (Q' yout,Frame P I (Q yout))) (tspec (R' yout,R yout))) ->
Triple (tspec (P',P)) (LiftTapes pM I) (fun y => tspec (R' y,R y)).
Proof.
intros ? ? <-%asPointwise. eapply LiftTapes_Spec. all:easy.
Qed.
Lemma LiftTapes_SpecT (sig F : finType)(m n : nat) (I : Vector.t (Fin.t n) m) P' (P : SpecV sig n) (k : nat) Q' (Q : F -> SpecV sig m) (pM : pTM sig^+ F m) :
dupfree I ->
TripleT (tspec (P',Downlift P I)) k pM (fun y => tspec (Q' y,Q y)) ->
TripleT (tspec (P',P)) k (LiftTapes pM I) (fun y => tspec (Q' y,Frame P I (Q y))).
Proof.
intros HDup (HTrip&HTrip').
split.
- apply LiftTapes_Spec; eauto.
- eapply TerminatesIn_monotone.
+ apply LiftTapes_Terminates; eauto.
+ intros tin k' (H&H'). split; auto.
now apply tape_fulfill_Downlift_select.
Qed.
Lemma LiftTapes_SpecT_con (sig : finType) (F : finType) (m n : nat) (I : Vector.t (Fin.t n) m)
P' (P : SpecV sig n) Q' (Q : F -> SpecV sig m) R' (R : F -> SpecV sig n)
(k : nat) (pM : pTM sig^+ F m) :
dupfree I ->
TripleT (tspec (P',Downlift P I)) k pM (fun y => tspec (Q' y,Q y)) ->
(forall yout, Entails (tspec (Q' yout,Frame P I (Q yout))) (tspec (R' yout,R yout))) ->
TripleT (tspec (P',P)) k (LiftTapes pM I) (fun y => tspec (R' y,R y)).
Proof. eauto using ConsequenceT_post, LiftTapes_SpecT. Qed.
Lemma Downlift_withSpace (m n : nat) (sig : Type) (P : SpecV sig n) (I : Vector.t (Fin.t n) m) (ss : Vector.t nat n) :
Downlift (withSpace P ss) I = withSpace (Downlift P I) (select I ss).
Proof.
unfold withSpace, Downlift.
eapply VectorSpec.eq_nth_iff; intros ? ? ->.
simpl_vector. rewrite !select_nth. simpl_vector. reflexivity.
Qed.
Lemma tspec_Downlift_withSpace (m n : nat) (sig : Type) P' (P : SpecV sig n) (I : Vector.t (Fin.t n) m) (ss : Vector.t nat n):
Entails ≃≃( P', Downlift (sig:=sig) (m:=m) (n:=n) (withSpace P ss) I) ≃≃( P',withSpace (Downlift P I) (select I ss)).
Proof. rewrite Entails_iff. intros H. erewrite <- Downlift_withSpace; eauto. Qed.
Lemma Triple_Downlift_withSpace (m n : nat) (sig : finType) P' (P : SpecV sig n) (I : Vector.t (Fin.t n) m) (ss : Vector.t nat n)
(F : Type) (M : pTM sig^+ F m) (Q : F -> Assert sig^+ m) :
Triple (tspec (P',withSpace (Downlift P I) (select I ss))) M Q ->
Triple (tspec (P',Downlift (withSpace P ss) I)) M Q.
Proof. now rewrite <- tspec_Downlift_withSpace. Qed.
Lemma TripleT_Downlift_withSpace (m n : nat) (sig : finType) P' (P : SpecV sig n) (I : Vector.t (Fin.t n) m) (ss : Vector.t nat n)
(F : Type) (k : nat) (M : pTM sig^+ F m) (Q : F -> Assert sig^+ m) :
TripleT (tspec (P',withSpace (Downlift P I) (select I ss))) k M Q ->
TripleT (tspec (P',Downlift (withSpace P ss) I)) k M Q.
Proof. now rewrite <- tspec_Downlift_withSpace. Qed.
Instance dec_ex_fin (n : nat) (P : Fin.t n -> Prop) (decP: forall (i : Fin.t n), dec (P i)) : dec (exists (i : Fin.t n), P i).
Proof.
induction n.
- right. intros (i&?). destruct_fin i.
- decide (P Fin0).
+ left. eauto.
+ specialize (IHn (fun i => P (Fin.FS i))). spec_assert IHn as [IH|IH] by eauto.
* left. destruct IH as (i&IH). exists (Fin.FS i). eauto.
* right. intros (j&H). pose proof (fin_destruct_S j) as [(j'&->) | ->]; eauto.
Qed.
Lemma Frame_withSpace (m n : nat) (sig : Type) (P : SpecV sig n) (P' : SpecV sig m) (I : Vector.t (Fin.t n) m) (ss : Vector.t nat n) (ss' : Vector.t nat m) :
dupfree I ->
Frame (withSpace P ss) I (withSpace P' ss') = withSpace (Frame P I P') (fill I ss ss').
Proof.
intros Hdup. unfold Frame,withSpace.
eapply VectorSpec.eq_nth_iff; intros ? i ->.
simpl_vector.
decide (exists j, I[@j]=i) as [(j&Hj)|Hj].
+ erewrite !fill_correct_nth by eauto. now simpl_vector.
+ assert (not_index I i).
{ hnf. intros (k&<-) % vect_nth_In'. contradict Hj. eauto. }
erewrite !fill_not_index by eauto. now simpl_vector.
Qed.
Lemma tspec_Frame_withSpace'
(m n : nat) (I : Vector.t (Fin.t n) m):
dupfree I -> forall (sig : Type) Q (P : SpecV sig n) (P' : SpecV sig m) (ss : Vector.t nat n) (ss' : Vector.t nat m),
Entails ≃≃( Q , Frame (withSpace P ss) I (withSpace P' ss')) ≃≃( Q, withSpace (Frame P I P') (fill I ss ss')).
Proof. intros H1 **. erewrite <- Frame_withSpace; eauto. Qed.
Lemma LiftTapes_Spec_space (sig F : finType) (m n : nat) (I : Vector.t (Fin.t n) m) P' (P : SpecV sig n) Q' (Q : F -> SpecV sig m) (pM : pTM sig^+ F m)
(ss : Vector.t nat n) (ss' : Vector.t nat m) :
dupfree I ->
Triple (tspec (P',withSpace (Downlift P I) (select I ss))) pM (fun y => tspec (Q' y,withSpace (Q y) ss')) ->
Triple (tspec (P',withSpace P ss)) (LiftTapes pM I) (fun y => tspec (Q' y,withSpace (Frame P I (Q y)) (fill I ss ss'))).
Proof.
intros H1 H2. rewrite <- Downlift_withSpace in H2. apply LiftTapes_Spec in H2. setoid_rewrite tspec_Frame_withSpace' in H2. all:eauto.
Qed.
Lemma LiftTapes_SpecT_space (sig F : finType) (m n : nat) (I : Vector.t (Fin.t n) m) P' (P : SpecV sig n) (k : nat) Q' (Q : F -> SpecV sig m) (pM : pTM sig^+ F m)
(ss : Vector.t nat n) (ss' : Vector.t nat m) :
dupfree I ->
TripleT (tspec (P',withSpace (Downlift P I) (select I ss))) k pM (fun y => tspec (Q' y,withSpace (Q y) ss')) ->
TripleT (tspec (P',withSpace P ss)) k (LiftTapes pM I) (fun y => tspec (Q' y,withSpace (Frame P I (Q y)) (fill I ss ss'))).
Proof.
intros H1 H2. rewrite <- Downlift_withSpace in H2. apply LiftTapes_SpecT in H2. setoid_rewrite tspec_Frame_withSpace' in H2. all:eauto.
Qed.
Lemma LiftTapes_Spec_space_con (sig : finType) (F : finType) (m n : nat) (I : Vector.t (Fin.t n) m)
P' (P : SpecV sig n) Q' (Q : F -> SpecV sig m) R' (R : F -> SpecV sig n) (ss : Vector.t nat n) (ss' : Vector.t nat m) (ss'' : Vector.t nat n)
(pM : pTM sig^+ F m) :
dupfree I ->
Triple (tspec (P',withSpace (Downlift P I) (select I ss))) pM (fun y => tspec (Q' y,withSpace (Q y) ss')) ->
(forall yout, Entails (tspec (Q' yout,withSpace (Frame P I (Q yout)) (fill I ss ss'))) (tspec (R' yout,withSpace (R yout) ss''))) ->
Triple (tspec (P',withSpace P ss)) (LiftTapes pM I) (fun y => tspec (R' y,withSpace (R y) ss'')).
Proof.
intros H1 H2 <-%asPointwise. rewrite <- Downlift_withSpace in H2. apply LiftTapes_Spec in H2.
setoid_rewrite tspec_Frame_withSpace' in H2. all:easy.
Qed.
Lemma LiftTapes_SpecT_space_con (sig : finType) (F : finType) (m n : nat) (I : Vector.t (Fin.t n) m)
P' (P : SpecV sig n) Q' (Q : F -> SpecV sig m) R' (R : F -> SpecV sig n) (ss : Vector.t nat n) (ss' : Vector.t nat m) (ss'' : Vector.t nat n)
(k : nat) (pM : pTM sig^+ F m) :
dupfree I ->
TripleT (tspec (P',withSpace (Downlift P I) (select I ss))) k pM (fun y => tspec (Q' y,withSpace (Q y) ss')) ->
(forall yout, Entails (tspec (Q' yout,withSpace (Frame P I (Q yout)) (fill I ss ss'))) (tspec (R' yout,withSpace (R yout) ss''))) ->
TripleT (tspec (P',withSpace P ss)) k (LiftTapes pM I) (fun y => tspec (R' y,withSpace (R y) ss'')).
Proof.
intros H1 H2 <-%asPointwise. rewrite <- Downlift_withSpace in H2. apply LiftTapes_SpecT in H2.
setoid_rewrite tspec_Frame_withSpace' in H2. all:easy.
Qed.
Section AlphabetLifting.
Variable (sig tau : Type).
Variable (retr : Retract sig tau).
Definition LiftSpec_single (T : RegSpec sig) : RegSpec tau :=
match T with
| Contains r x => Contains (ComposeRetract retr r) x
| Contains_size r x s => Contains_size (ComposeRetract retr r) x s
| Void => Void
| Void_size s => Void_size s
| Custom p => Custom (fun t => p (surjectTape (Retr_g) (inl UNKNOWN) t))
end.
Variable (n : nat).
Definition LiftSpec (T : SpecV sig n) : SpecV tau n :=
Vector.map LiftSpec_single T.
Lemma LiftSpec_surjectTape_tspec_single t T :
tspec_single (LiftSpec_single T) t ->
tspec_single T (surjectTape Retr_g (inl UNKNOWN) t).
Proof. destruct T; cbn in *; intros; simpl_surject; eauto. Qed.
Lemma LiftSpec_surjectTape_tspec_single' t T :
tspec_single T (surjectTape Retr_g (inl UNKNOWN) t) ->
tspec_single (LiftSpec_single T) t.
Proof. destruct T; cbn in *; intros; simpl_surject; eauto. Qed.
Lemma LiftSpec_surjectTapes_tspec tin P' P :
tin ≃≃ (P', LiftSpec P) ->
surjectTapes Retr_g (inl UNKNOWN) tin ≃≃ (P',P).
Proof.
intros (H'&H)%tspecE. eapply tspecI. easy.
intros i; specialize (H i); cbn. unfold LiftSpec in *.
simpl_tape in *. now apply LiftSpec_surjectTape_tspec_single.
Qed.
Lemma LiftSpec_surjectTapes_tspec' tin P P':
surjectTapes Retr_g (inl UNKNOWN) tin ≃≃ (P',P) ->
tin ≃≃ (P',LiftSpec P).
Proof.
intros (H'&H)%tspecE. eapply tspecI. easy. unfold LiftSpec in *.
intros i; specialize (H i); cbn.
simpl_tape in *. now apply LiftSpec_surjectTape_tspec_single'.
Qed.
End AlphabetLifting.
Lemma LiftSpec_withSpace_single (sig tau : Type) (I : Retract sig tau) (P : RegSpec sig) (s : nat) :
LiftSpec_single I (withSpace_single P s) = withSpace_single (LiftSpec_single I P) s.
Proof. destruct P; cbn; eauto. Qed.
Lemma LiftSpec_withSpace (sig tau : Type) (n : nat) (I : Retract sig tau) (P : SpecV sig n) (ss : Vector.t nat n) :
LiftSpec I (withSpace P ss) = withSpace (LiftSpec I P) ss.
Proof.
eapply VectorSpec.eq_nth_iff; intros ? ? ->. unfold LiftSpec, withSpace.
simpl_vector. apply LiftSpec_withSpace_single.
Qed.
Section AlphabetLifting'.
Variable (sig tau : finType) (n : nat).
Variable (retr : Retract sig tau).
Lemma ChangeAlphabet_Spec_ex (F : finType) X P' (Ctx : X -> Prop -> Prop) (P : SpecV sig n) (pM : pTM sig^+ F n) Q' (Q : X -> F -> SpecV sig n) :
Triple (tspec (P',P)) pM (fun y t => exists x:X, Ctx x (tspec (Q' x y,Q x y) t)) ->
(forall x, Proper (Basics.impl ==> Basics.impl) (Ctx x)) ->
Triple (tspec (P',LiftSpec retr P)) (ChangeAlphabet pM retr)
(fun yout t => exists x, Ctx x (tspec (Q' x yout,LiftSpec retr (Q x yout)) t)).
Proof.
rewrite !Triple_iff. intros HTrip HCtx. eapply Realise_monotone.
- TM_Correct. eassumption.
- intros tin (yout, tout) H Henc. cbn in *.
spec_assert H by now apply LiftSpec_surjectTapes_tspec.
destruct H as (x&H). exists x. cbv in HCtx. eapply HCtx. 2:apply H.
now apply LiftSpec_surjectTapes_tspec'.
Qed.
Lemma ChangeAlphabet_Spec (F : finType) P' (P : SpecV sig n) (pM : pTM sig^+ F n) Q' (Q : F -> SpecV sig n) :
Triple (tspec (P',P)) pM (fun yout => tspec (Q' yout,Q yout)) ->
Triple (tspec (P',LiftSpec retr P)) (ChangeAlphabet pM retr) (fun yout => tspec (Q' yout,LiftSpec retr (Q yout))).
Proof.
rewrite !Triple_iff. intros HTrip. eapply Realise_monotone.
- TM_Correct. eassumption.
- intros tin (yout, tout) H Henc. cbn in *.
spec_assert H by now apply LiftSpec_surjectTapes_tspec.
now apply LiftSpec_surjectTapes_tspec'.
Qed.
Lemma ChangeAlphabet_SpecT (F : finType) P' (P : SpecV sig n) (k : nat) (pM : pTM sig^+ F n) Q' (Q : F -> SpecV sig n) :
TripleT (tspec (P',P)) k pM (fun yout => tspec (Q' yout, Q yout)) ->
TripleT (tspec (P',LiftSpec retr P)) k (ChangeAlphabet pM retr) (fun yout => tspec (Q' yout,LiftSpec retr (Q yout))).
Proof.
intros HTrip. split.
{ apply ChangeAlphabet_Spec. eapply TripleT_Triple; eauto. }
{
eapply TerminatesIn_monotone.
- TM_Correct. apply HTrip.
- unfold Triple_TRel. intros tin k' (H&Hk). cbn. split; auto.
now apply LiftSpec_surjectTapes_tspec.
}
Qed.
Lemma ChangeAlphabet_Spec_pre_post (F : finType)
P0 (P : SpecV sig n) (P' : SpecV tau n)
(pM : pTM sig^+ F n)
Q0 (Q : F -> SpecV sig n) (Q' : F -> SpecV tau n) :
Triple (tspec (P0,P)) pM (fun yout => tspec (Q0 yout, Q yout) ) ->
(Entails ≃≃( P0,P') ≃≃( P0,LiftSpec retr P)) ->
(forall yout, Entails ≃≃( Q0 yout, LiftSpec retr (Q yout)) ≃≃( (Q0 yout,Q' yout))) ->
Triple (tspec (P0, P')) (ChangeAlphabet pM retr) (fun yout => tspec (Q0 yout, Q' yout)).
Proof.
intros H1 H2 H3.
eapply Consequence.
- apply ChangeAlphabet_Spec. apply H1.
- apply H2.
- apply H3.
Qed.
Lemma ChangeAlphabet_SpecT_pre_post (F : finType)
P0 (P : SpecV sig n) (P' : SpecV tau n)
(k : nat) (pM : pTM sig^+ F n)
Q0 (Q : F -> SpecV sig n) (Q' : F -> SpecV tau n) :
TripleT (tspec (P0,P)) k pM (fun yout => tspec (Q0 yout, Q yout) ) ->
(Entails ≃≃( P0,P') ≃≃( P0,LiftSpec retr P)) ->
(forall yout, Entails ≃≃( Q0 yout, LiftSpec retr (Q yout)) ≃≃( (Q0 yout,Q' yout))) ->
TripleT (tspec (P0, P')) k (ChangeAlphabet pM retr) (fun yout => tspec (Q0 yout, Q' yout)).
Proof.
intros H1 H2 H3.
eapply ConsequenceT.
- apply ChangeAlphabet_SpecT. apply H1.
- apply H2.
- apply H3.
- reflexivity.
Qed.
Lemma ChangeAlphabet_Spec_pre (F : finType)
P0 (P : SpecV sig n) (P' : SpecV tau n)
(pM : pTM sig^+ F n)
Q0 (Q : F -> SpecV sig n) :
Triple (tspec (P0,P)) pM (fun yout => tspec (Q0 yout, Q yout)) ->
(Entails ≃≃( P0,P') ≃≃( P0, LiftSpec retr P)) ->
Triple (tspec (P0,P')) (ChangeAlphabet pM retr) (fun yout => tspec (Q0 yout,LiftSpec retr (Q yout))).
Proof.
intros H1 H2.
eapply Consequence.
- apply ChangeAlphabet_Spec. apply H1.
- apply H2.
- eauto.
Qed.
Lemma ChangeAlphabet_SpecT_pre (F : finType)
P0 (P : SpecV sig n) (P' : SpecV tau n)
(k : nat) (pM : pTM sig^+ F n)
Q0 (Q : F -> SpecV sig n) :
TripleT (tspec (P0,P)) k pM (fun yout => tspec (Q0 yout, Q yout)) ->
(Entails ≃≃( P0, P') ≃≃( P0,LiftSpec retr P)) ->
TripleT (tspec (P0,P')) k (ChangeAlphabet pM retr) (fun yout => tspec (Q0 yout,LiftSpec retr (Q yout))).
Proof.
intros H1 H2.
eapply ConsequenceT.
- apply ChangeAlphabet_SpecT. apply H1.
- apply H2.
- eauto.
- reflexivity.
Qed.
Lemma ChangeAlphabet_Spec_space_pre_post (F : finType)
P0 (P : SpecV sig n) (P' : SpecV tau n)
(pM : pTM sig^+ F n)
Q0 (Q : F -> SpecV sig n) (Q' : F -> SpecV tau n)
(ss ss' : Vector.t nat n) :
Triple (tspec (P0,withSpace P ss)) pM (fun yout => tspec (Q0 yout,withSpace (Q yout) ss')) ->
(Entails ≃≃( P0, withSpace P' ss) ≃≃( P0, withSpace (LiftSpec retr P) ss)) ->
(forall yout, Entails ≃≃( Q0 yout, withSpace (LiftSpec retr (Q yout)) ss') (tspec (Q0 yout,withSpace (Q' yout) ss'))) ->
Triple (tspec (P0,withSpace P' ss)) (ChangeAlphabet pM retr) (fun yout => tspec (Q0 yout, withSpace (Q' yout) ss')).
Proof.
intros H1 H2 H3.
eapply Consequence.
- apply ChangeAlphabet_Spec. apply H1.
- rewrite LiftSpec_withSpace. apply H2.
- setoid_rewrite Entails_iff. cbn. intros. rewrite LiftSpec_withSpace in H. now apply H3.
Qed.
Lemma ChangeAlphabet_SpecT_space_pre_post (F : finType)
P0 (P : SpecV sig n) (P' : SpecV tau n)
(k : nat) (pM : pTM sig^+ F n)
Q0 (Q : F -> SpecV sig n) (Q' : F -> SpecV tau n)
(ss ss' : Vector.t nat n) :
TripleT (tspec (P0,withSpace P ss)) k pM (fun yout => tspec (Q0 yout,withSpace (Q yout) ss')) ->
(Entails ≃≃( P0, withSpace P' ss) ≃≃( P0, withSpace (LiftSpec retr P) ss)) ->
(forall yout, Entails ≃≃( Q0 yout, withSpace (LiftSpec retr (Q yout)) ss') (tspec (Q0 yout,withSpace (Q' yout) ss'))) ->
TripleT (tspec (P0,withSpace P' ss)) k (ChangeAlphabet pM retr) (fun yout => tspec (Q0 yout, withSpace (Q' yout) ss')).
Proof.
intros H1 H2 H3.
eapply ConsequenceT.
- apply ChangeAlphabet_SpecT. apply H1.
- rewrite LiftSpec_withSpace. apply H2.
- setoid_rewrite Entails_iff. cbn. intros. rewrite LiftSpec_withSpace in H. now apply H3.
- reflexivity.
Qed.
Lemma ChangeAlphabet_Spec_space_pre (F : finType)
P0 (P : SpecV sig n) (P' : SpecV tau n)
(pM : pTM sig^+ F n)
Q0 (Q : F -> SpecV sig n)
(ss ss' : Vector.t nat n) :
Triple (tspec (P0,withSpace P ss)) pM (fun yout => tspec (Q0 yout,withSpace (Q yout) ss')) ->
Entails ≃≃( P0, withSpace P' ss) ≃≃( P0, withSpace (LiftSpec retr P) ss) ->
Triple (tspec (P0, withSpace P' ss)) (ChangeAlphabet pM retr) (fun yout => tspec (Q0 yout, withSpace (LiftSpec retr (Q yout)) ss')).
Proof.
intros H1 H2.
eapply Consequence.
- apply ChangeAlphabet_Spec. apply H1.
- rewrite LiftSpec_withSpace. apply H2.
- setoid_rewrite Entails_iff. cbn. intros. rewrite LiftSpec_withSpace in H. now apply H.
Qed.
Lemma ChangeAlphabet_SpecT_space_pre (F : finType)
P0 (P : SpecV sig n) (P' : SpecV tau n)
(k : nat) (pM : pTM sig^+ F n)
Q0 (Q : F -> SpecV sig n)
(ss ss' : Vector.t nat n) :
TripleT (tspec (P0,withSpace P ss)) k pM (fun yout => tspec (Q0 yout, withSpace (Q yout) ss')) ->
Entails ≃≃( P0, withSpace P' ss) ≃≃( P0, withSpace (LiftSpec retr P) ss) ->
TripleT (tspec (P0, withSpace P' ss)) k (ChangeAlphabet pM retr) (fun yout => tspec (Q0 yout,withSpace (LiftSpec retr (Q yout)) ss')).
Proof.
intros H1 H2.
eapply ConsequenceT.
- apply ChangeAlphabet_SpecT. apply H1.
- rewrite LiftSpec_withSpace. apply H2.
- setoid_rewrite Entails_iff. cbn. intros. rewrite LiftSpec_withSpace in H. now apply H.
- reflexivity.
Qed.
End AlphabetLifting'.
Global Arguments withSpace : simpl never.
Notation "'SpecFalse'" := ([False],_): spec_scope.
From Undecidability.TM Require Import TMTac.
From Undecidability.TM Require Export CodeTM LiftTapes ChangeAlphabet.
Section RegSpec.
Variable sig : Type.
Local Notation "sig '^+'" := ((boundary + sig) % type) (at level 0) : type_scope.
Inductive RegSpec : Type :=
| Contains {sigX X : Type} {cX : codable sigX X} (r : Retract sigX sig) : X -> RegSpec
| Contains_size {sigX X : Type} {cX : codable sigX X} (r : Retract sigX sig) : X -> nat -> RegSpec
| Void : RegSpec
| Void_size : nat -> RegSpec
| Custom : (tape sig^+ -> Prop) -> RegSpec.
Definition tspec_single (spec : RegSpec) (t : tape sig^+) : Prop :=
match spec with
| Contains r x => t ≃(r) x
| Contains_size r x s => t ≃(r; s) x
| Void => isVoid t
| Void_size s => isVoid_size t s
| Custom p => p t
end.
Lemma tspec_single_Contains_size_Contains {sigX X : Type} {cX : codable sigX X} (r : Retract sigX sig) (x : X) (s : nat) (t : tape sig^+) :
tspec_single (Contains_size r x s) t -> tspec_single (Contains r x) t.
Proof. cbn. auto. Qed.
Definition SpecV n := Vector.t (RegSpec) n.
Definition Spec n :Type := list Prop * SpecV n.
Definition tspec {n : nat} (spec : Spec n) : Assert sig^+ n :=
match spec with
| (P,spec) => fun (t : tapes sig^+ n) =>
List.fold_right and (forall (i : Fin.t n), tspec_single spec[@i] t[@i]) P
end.
Arguments tspec : simpl never.
Lemma tspecE {n : nat} Ps Pv t:
tspec (n:=n) (Ps,Pv) t -> (List.fold_right and True Ps /\ forall (i : Fin.t n), tspec_single Pv[@i] t[@i]).
Proof.
cbn. induction Ps;cbn in *. easy. intros []. specialize (IHPs H0) as [H' ?]. split. all:eauto.
Qed.
Lemma tspecI {n : nat} P t:
(List.fold_right and True (fst P))
-> (forall (i : Fin.t n), tspec_single (snd P)[@i] t[@i])
-> tspec (n:=n) P t.
Proof.
destruct P as (Ps&P). induction Ps;cbn in *. easy. intros H' ?. split. now eapply H';left.
eapply IHPs. all:easy.
Qed.
Lemma tspec_iff {n : nat} Ps Pv t:
tspec (n:=n) (Ps,Pv) t <-> (List.fold_right and True Ps /\ forall (i : Fin.t n), tspec_single Pv[@i] t[@i]).
Proof. split. eapply tspecE;eauto. intros []. eapply tspecI;eauto. Qed.
Lemma tspec_Entails {n : nat} Ps Pv:
Entails (tspec (n:=n) (Ps,Pv)) (fun t => List.fold_right and True Ps /\ forall (i : Fin.t n), tspec_single Pv[@i] t[@i]).
Proof. apply EntailsI. now setoid_rewrite tspec_iff. Qed.
Definition withSpace_single (P : RegSpec) (size : nat) :=
match P with
| Contains r x => Contains_size r x size
| Void => Void_size size
| _ => P
end.
Definition withSpace {n : nat} (P : SpecV n) (spaces : Vector.t nat n) : SpecV n :=
Vector.map2 withSpace_single P spaces.
Lemma tspec_single_withSpace_tspec_single (P : RegSpec) (size : nat) t :
tspec_single (withSpace_single P size) t -> tspec_single P t.
Proof. intros. destruct P; cbn in *; auto. Qed.
Lemma tspec_withSpace_tspec {n : nat} Q P (s : Vector.t nat n) t :
tspec (Q,withSpace P s) t -> tspec (Q,P) t.
Proof. unfold withSpace. intros [HP H]%tspecE. apply tspecI. easy. intros i; specialize (H i). simpl_vector in *. eapply tspec_single_withSpace_tspec_single; eauto. Qed.
Definition dummy_size (t : tape sig^+) (P : RegSpec) : nat :=
match P with
| Contains r x => length (left t)
| Void => length (tape_local_l t)
| _ => 0
end.
Lemma tspec_single_tspec_single_withSpace (P : RegSpec) t :
tspec_single P t -> tspec_single (withSpace_single P (dummy_size t P)) t.
Proof. intros H. destruct P; cbn in *; eauto. Qed.
Definition dummy_sizes (n : nat) (t : tapes sig^+ n) (P : SpecV n) : Vector.t nat n :=
Vector.map2 dummy_size t P.
Lemma tspec_tspec_withSpace (n : nat) Q (P : SpecV n) t :
tspec (Q,P) t -> tspec (Q,withSpace P (dummy_sizes t P)) t.
Proof.
unfold withSpace, dummy_sizes in *. intros [HP ?]%tspecE. eapply tspecI. cbn in *; auto.
intros i; specialize (H i); cbn in *. simpl_vector. now apply tspec_single_tspec_single_withSpace.
Qed.
End RegSpec.
Arguments Custom {sig}.
Arguments Void {sig}.
Arguments Void_size {sig}.
Arguments dummy_sizes : simpl never.
Hint Resolve tspec_single_Contains_size_Contains : core.
Declare Scope spec_scope.
Delimit Scope spec_scope with spec.
Bind Scope spec_scope with Spec.
Notation "'≃≃(' S ')'" := (tspec S%spec) (at level 0, S at level 200, no associativity, format "'≃≃(' S ')'").
Notation "'≃≃(' P ',' S ')'" := (tspec (P%list,S%vector)) (at level 0, P at level 200, S at level 200, no associativity, format "'≃≃(' P ',' S ')'").
Notation "t ≃≃ S" := (tspec S%spec t) (at level 70, no associativity).
Notation "≃( I ) x" := (Contains I x) (at level 70, I at level 200, no associativity, format "'≃(' I ')' x").
Notation "≃( I ';' s ) x" := (Contains_size I s x) (at level 70, I at level 200, s at level 200, no associativity, format "'≃(' I ';' s ')' x").
Arguments tspec _%spec _.
Fixpoint implList (Ps : list Prop) (Q : Prop) :=
match Ps with
[] => Q
| P::Ps => P -> implList Ps Q
end.
Arguments implList !_ _.
Instance fold_right_impl' : Proper (Forall2 Basics.impl --> Basics.impl ==> Basics.impl) (implList).
Proof. intros xs;induction xs;cbn;intros ? H';inv H';cbn. easy. firstorder. Qed.
Instance fold_right_iff : Proper (Forall2 iff ==> iff ==> iff) (implList).
Proof. intros xs;induction xs;cbn;intros ? H';inv H';cbn. easy. firstorder. Qed.
Lemma implList_iff P (Ps : list Prop):
implList Ps P
<-> (List.fold_right and True Ps -> P).
Proof.
induction Ps in P|-*;cbn. firstorder. setoid_rewrite IHPs. tauto.
Qed.
Lemma implListE P (Ps : list Prop): implList Ps P -> (List.fold_right and True Ps -> P).
Proof. now rewrite implList_iff. Qed.
Lemma implListI (P:Prop) (Ps : list Prop): (List.fold_right and True Ps -> P) -> implList Ps P.
Proof. now rewrite implList_iff. Qed.
Instance Forall2_refl X (R: X -> _): Reflexive R -> Reflexive (Forall2 R).
Proof. intros ? xs;induction xs;eauto. Qed.
Lemma tspec_introPure (sig: finType) (n : nat) P (Ps : SpecV sig n) Q:
implList P (Entails (≃≃([],Ps)) Q)
-> Entails (tspec (P,Ps)) Q.
Proof.
setoid_rewrite Entails_iff. rewrite implList_iff. intros H ? []%tspecE. eapply H. eassumption. now apply tspecI.
Qed.
Lemma tspec_revertPure (sig: finType) (n : nat) (P0:Prop) P (Ps : SpecV sig n) Q:
P0
-> Entails (tspec (P0::P,Ps)) Q
-> Entails (tspec (P,Ps)) Q.
Proof.
setoid_rewrite Entails_iff. unfold tspec;cbn. intuition.
Qed.
Lemma Triple_introPure (F sig: finType) (n : nat) P (Ps : SpecV sig n) Q (pM : pTM sig^+ F n) :
implList P (Triple (≃≃([],Ps)) pM Q)
-> Triple (tspec (P,Ps)) pM Q.
Proof.
intros. rewrite tspec_Entails. apply Triple_and_pre. cbn in H. now rewrite <- implList_iff.
Qed.
Lemma TripleT_introPure (sig F : finType) (n : nat) P (Ps : SpecV sig n) Q k (pM : pTM sig^+ F n) :
implList P (TripleT (≃≃([],Ps)) k pM Q)
-> TripleT (tspec (P,Ps)) k pM Q.
Proof.
intros. rewrite tspec_Entails. apply TripleT_and_pre. cbn in H. now rewrite <- implList_iff.
Qed.
Definition SpecVTrue {sig : Type} {n : nat} : SpecV sig n := Vector.const (Custom (fun _ => True)) n.
Arguments tspec : simpl never.
Definition appSize {n : nat} : Vector.t (nat->nat) n -> Vector.t nat n -> Vector.t nat n :=
fun fs s => tabulate (fun i => fs[@i] s[@i]).
Lemma Triple_RemoveSpace_ex (n : nat) (sig : finType) (F : Type) X
(P : SpecV sig n) P' (M : pTM sig^+ F n) Q Q' Ctx (fs : _ -> Vector.t (nat->nat) n) :
(forall s, Triple (tspec (P',withSpace P s)) M (fun y t => exists x:X, Ctx x (tspec (Q' x y,withSpace (Q x y) (appSize (fs x) s)) t))) ->
(forall x, Proper (Basics.impl ==> Basics.impl) (Ctx x)) ->
Triple (tspec (P',P)) M (fun y t => exists x, Ctx x (tspec (Q' x y ,Q x y) t)).
Proof.
intros HTrip Hctx. setoid_rewrite Triple_iff in HTrip. rewrite Triple_iff.
eapply Realise_monotone with
(R' := fun tin '(yout, tout) => forall s, tspec (P',withSpace P s) tin
-> exists x:X, Ctx x (tspec (Q' x yout,withSpace (Q x yout) (appSize (fs x) s)) tout)).
- unfold Triple_Rel, Realise in *. intros tin k outc HLoop. intros s HP.
specialize HTrip with (1 := HLoop) (2 := HP) as [x H'']. eexists. eauto.
- clear HTrip. intros tin (yout, tout). intros H HP.
specialize (H (dummy_sizes tin P)). spec_assert H by now apply tspec_tspec_withSpace.
destruct H as (x&H). exists x. eapply Hctx. 2:eassumption.
intro H'. now apply tspec_withSpace_tspec in H'.
Qed.
Lemma Triple_RemoveSpace (n : nat) (sig : finType) (F : Type) (P : SpecV sig n) P' (M : pTM sig^+ F n) (Q : F -> SpecV sig n) Q' (fs : Vector.t (nat->nat) n) :
(forall s, Triple (tspec (P',withSpace P s)) M (fun y => tspec (Q' y,withSpace (Q y) (appSize fs s)))) ->
Triple (tspec (P',P)) M (fun y => tspec (Q' y ,Q y)).
Proof.
intro HTrip. setoid_rewrite Triple_iff in HTrip. rewrite Triple_iff.
eapply Realise_monotone with (R' := fun tin '(yout, tout) => forall s, tspec (P',withSpace P s) tin -> tspec (Q' yout,withSpace (Q yout) (appSize fs s)) tout).
- unfold Triple_Rel, Realise in *. intros tin k outc HLoop. intros s HP.
now specialize HTrip with (1 := HLoop) (2 := HP).
- clear HTrip. intros tin (yout, tout). intros H HP.
specialize (H (dummy_sizes tin P)). spec_assert H by now apply tspec_tspec_withSpace.
now apply tspec_withSpace_tspec in H.
Qed.
Lemma TripleT_RemoveSpace (n : nat) (sig : finType) (F : Type) P' (P : SpecV sig n) (k : nat) (M : pTM sig^+ F n) Q' (Q : F -> SpecV sig n) (fs : Vector.t (nat->nat) n) :
(forall s, TripleT (tspec (P',withSpace P s)) k M (fun y => tspec (Q' y,withSpace (Q y) (appSize fs s)))) ->
TripleT (tspec (P',P)) k M (fun y => tspec (Q' y,Q y)).
Proof.
intros HTrip. split.
- eapply Triple_RemoveSpace. intros s. apply HTrip.
- setoid_rewrite TripleT_iff in HTrip.
eapply TerminatesIn_monotone with (T' := fun tin k' => tspec (P',P) tin /\ k <= k').
+ unfold Triple_TRel, TerminatesIn in *. intros tin k' (HP&Hk).
specialize (HTrip (dummy_sizes tin P)) as (_&HT).
specialize HT with (tin0 := tin) (k0 := k). spec_assert HT as (conf&HLoop).
{ split. now apply tspec_tspec_withSpace. reflexivity. }
exists conf. eapply loop_monotone; eauto.
+ unfold Triple_TRel. intros tin k' (HP&Hk). eauto.
Qed.
Instance fold_right_and : Proper (iff ==> Forall2 iff ==> iff) (fold_right and).
Proof. intros ? ? ? xs;induction xs;cbn;intros ? H';inv H';cbn. easy. firstorder. Qed.
Instance fold_right_and' : Proper (Basics.impl ==> Forall2 iff ==> Basics.impl) (fold_right and).
Proof. intros ? ? ? xs;induction xs;cbn;intros ? H';inv H';cbn. easy. firstorder. Qed.
Lemma tspec_solve (sig : Type) (n : nat) (t : tapes (boundary+sig) n) (R : SpecV sig n) P:
List.fold_right and (forall i, tspec_single R[@i] t[@i]) P ->
tspec (P,R) t.
Proof. refine (fun P => P). Qed.
Lemma tspec_space_solve (sig : Type) (n : nat) (t : tapes (boundary+sig) n) (R : SpecV sig n) P (ss : Vector.t nat n) :
List.fold_right and (forall i, tspec_single (withSpace_single R[@i] ss[@i]) t[@i]) P ->
tspec (P,withSpace R ss) t.
Proof. unfold withSpace. intros. apply tspec_solve. simpl_vector. auto. Qed.
Lemma tspec_ext (sig : finType) (n : nat) (t : tapes (boundary+sig) n) (P P' : list Prop) (R R' : Vector.t (RegSpec sig) n) :
tspec (P',R') t ->
implList P' (List.fold_right and True P) ->
(forall i, tspec_single R'[@i] t[@i] -> tspec_single R[@i] t[@i]) ->
tspec (P,R) t.
Proof.
intros [HP H1]%tspecE H1' H2. eapply tspecI.
eapply implList_iff. 2:eassumption. eapply fold_right_impl'. 2:reflexivity. 2:eassumption. easy.
intros i; specialize (H1 i); specialize (H2 i); eauto.
Qed.
Lemma tspec_space_ext (sig : finType) (n : nat) (t : tapes (boundary+sig) n) (P P':list Prop) (R R' : SpecV sig n)
(ss ss' : Vector.t nat n) :
tspec (P',withSpace R' ss') t ->
implList P' (List.fold_right and True P) ->
(forall i, tspec_single (withSpace_single R'[@i] ss'[@i]) t[@i] -> tspec_single (withSpace_single R[@i] ss[@i]) t[@i]) ->
tspec (P,withSpace R ss) t.
Proof.
unfold withSpace. intros [HP H1]%tspecE H1' H2. eapply tspecI.
eapply implList_iff. 2:eassumption. eapply fold_right_impl'. 2:reflexivity. 2:eassumption. easy.
intros i; specialize (H1 i); specialize (H2 i); eauto.
cbn. simpl_vector in *; cbn. eauto.
Qed.
Section Lifting.
Variable (sig : Type).
Variable (m n : nat).
Variable (P : @SpecV sig n).
Variable (I : Vector.t (Fin.t n) m). Hypothesis (HI : dupfree I).
Definition Downlift : @SpecV sig m :=
(select I P).
Lemma tape_fulfill_Downlift_select P' tp :
tspec (P',P) tp ->
tspec (P',Downlift) (select I tp).
Proof.
unfold Downlift.
intros [? H]%tspecE.
eapply tspecI. easy.
intros i;cbn. rewrite !select_nth. easy.
Qed.
Definition Frame (Q : @SpecV sig m) : @SpecV sig n := fill I P Q.
End Lifting.
Lemma LiftTapes_Spec_ex (sig : finType) X (F : finType) (m n : nat) (I : Vector.t (Fin.t n) m)
P' P Q' Q (pM : pTM sig^+ F m) :
dupfree I ->
Triple (tspec (P',Downlift P I)) pM (fun y t => exists x:X, tspec (Q' x y,Q x y) t) ->
Triple (tspec (P',P)) (LiftTapes pM I) (fun y t=> exists x, tspec (Q' x y,Frame P I (Q x y)) t ).
Proof.
unfold Frame. rewrite !Triple_iff.
intros HDup HTrip.
eapply Realise_monotone.
{ apply LiftTapes_Realise. assumption. apply HTrip. }
{
intros tin (yout, tout) (H&HInj). cbn -[Downlift tspec] in *.
intros HP.
spec_assert H by now apply tape_fulfill_Downlift_select.
destruct H as [x H].
eapply tspecE in H as [H' H]. eapply tspecE in HP as [HP' HP].
exists x.
eapply tspecI;cbn.
{ clear - H' HP'. induction P';cbn in *. all:firstorder. }
clear H' HP'.
hnf. intros j. decide (Vector.In j I) as [HD|HD].
- unfold Frame.
apply vect_nth_In' in HD as (ij&HD).
erewrite fill_correct_nth; eauto.
specialize (H ij).
now rewrite select_nth, HD in H.
- unfold Frame. rewrite fill_not_index; eauto.
specialize (HInj j HD). rewrite HInj. now apply HP.
}
Qed.
Lemma LiftTapes_Spec (sig : finType) (F : finType) (m n : nat) (I : Vector.t (Fin.t n) m) P' (P : SpecV sig n) Q' (Q : F -> SpecV sig m) (pM : pTM sig^+ F m) :
dupfree I ->
Triple (tspec (P',Downlift P I)) pM (fun y => tspec (Q' y,Q y)) ->
Triple (tspec (P',P)) (LiftTapes pM I) (fun y => tspec (Q' y,Frame P I (Q y))).
Proof.
unfold Frame. rewrite !Triple_iff.
intros HDup HTrip.
eapply Realise_monotone.
{ apply LiftTapes_Realise. assumption. apply HTrip. }
{
intros tin (yout, tout) (H&HInj). cbn -[Downlift tspec] in *.
intros HP.
spec_assert H by now apply tape_fulfill_Downlift_select.
eapply tspecE in H as [H' H]. eapply tspecE in HP as [HP' HP].
eapply tspecI;cbn.
{ clear - H' HP'. induction P';cbn in *. all:firstorder. }
clear H' HP'.
hnf. intros j. decide (Vector.In j I) as [HD|HD].
- unfold Frame.
apply vect_nth_In' in HD as (ij&HD).
erewrite fill_correct_nth; eauto.
specialize (H ij).
now rewrite select_nth, HD in H.
- unfold Frame. rewrite fill_not_index; eauto.
specialize (HInj j HD). rewrite HInj. now apply HP.
}
Qed.
Lemma LiftTapes_Spec_con (sig : finType) (F : finType) (m n : nat) (I : Vector.t (Fin.t n) m) P' (P : SpecV sig n) Q' (Q : F -> SpecV sig m) R' (R : F -> SpecV sig n) (pM : pTM sig^+ F m) :
dupfree I ->
Triple (tspec (P',Downlift P I)) pM (fun y => tspec (Q' y,Q y)) ->
(forall yout, Entails (tspec (Q' yout,Frame P I (Q yout))) (tspec (R' yout,R yout))) ->
Triple (tspec (P',P)) (LiftTapes pM I) (fun y => tspec (R' y,R y)).
Proof.
intros ? ? <-%asPointwise. eapply LiftTapes_Spec. all:easy.
Qed.
Lemma LiftTapes_SpecT (sig F : finType)(m n : nat) (I : Vector.t (Fin.t n) m) P' (P : SpecV sig n) (k : nat) Q' (Q : F -> SpecV sig m) (pM : pTM sig^+ F m) :
dupfree I ->
TripleT (tspec (P',Downlift P I)) k pM (fun y => tspec (Q' y,Q y)) ->
TripleT (tspec (P',P)) k (LiftTapes pM I) (fun y => tspec (Q' y,Frame P I (Q y))).
Proof.
intros HDup (HTrip&HTrip').
split.
- apply LiftTapes_Spec; eauto.
- eapply TerminatesIn_monotone.
+ apply LiftTapes_Terminates; eauto.
+ intros tin k' (H&H'). split; auto.
now apply tape_fulfill_Downlift_select.
Qed.
Lemma LiftTapes_SpecT_con (sig : finType) (F : finType) (m n : nat) (I : Vector.t (Fin.t n) m)
P' (P : SpecV sig n) Q' (Q : F -> SpecV sig m) R' (R : F -> SpecV sig n)
(k : nat) (pM : pTM sig^+ F m) :
dupfree I ->
TripleT (tspec (P',Downlift P I)) k pM (fun y => tspec (Q' y,Q y)) ->
(forall yout, Entails (tspec (Q' yout,Frame P I (Q yout))) (tspec (R' yout,R yout))) ->
TripleT (tspec (P',P)) k (LiftTapes pM I) (fun y => tspec (R' y,R y)).
Proof. eauto using ConsequenceT_post, LiftTapes_SpecT. Qed.
Lemma Downlift_withSpace (m n : nat) (sig : Type) (P : SpecV sig n) (I : Vector.t (Fin.t n) m) (ss : Vector.t nat n) :
Downlift (withSpace P ss) I = withSpace (Downlift P I) (select I ss).
Proof.
unfold withSpace, Downlift.
eapply VectorSpec.eq_nth_iff; intros ? ? ->.
simpl_vector. rewrite !select_nth. simpl_vector. reflexivity.
Qed.
Lemma tspec_Downlift_withSpace (m n : nat) (sig : Type) P' (P : SpecV sig n) (I : Vector.t (Fin.t n) m) (ss : Vector.t nat n):
Entails ≃≃( P', Downlift (sig:=sig) (m:=m) (n:=n) (withSpace P ss) I) ≃≃( P',withSpace (Downlift P I) (select I ss)).
Proof. rewrite Entails_iff. intros H. erewrite <- Downlift_withSpace; eauto. Qed.
Lemma Triple_Downlift_withSpace (m n : nat) (sig : finType) P' (P : SpecV sig n) (I : Vector.t (Fin.t n) m) (ss : Vector.t nat n)
(F : Type) (M : pTM sig^+ F m) (Q : F -> Assert sig^+ m) :
Triple (tspec (P',withSpace (Downlift P I) (select I ss))) M Q ->
Triple (tspec (P',Downlift (withSpace P ss) I)) M Q.
Proof. now rewrite <- tspec_Downlift_withSpace. Qed.
Lemma TripleT_Downlift_withSpace (m n : nat) (sig : finType) P' (P : SpecV sig n) (I : Vector.t (Fin.t n) m) (ss : Vector.t nat n)
(F : Type) (k : nat) (M : pTM sig^+ F m) (Q : F -> Assert sig^+ m) :
TripleT (tspec (P',withSpace (Downlift P I) (select I ss))) k M Q ->
TripleT (tspec (P',Downlift (withSpace P ss) I)) k M Q.
Proof. now rewrite <- tspec_Downlift_withSpace. Qed.
Instance dec_ex_fin (n : nat) (P : Fin.t n -> Prop) (decP: forall (i : Fin.t n), dec (P i)) : dec (exists (i : Fin.t n), P i).
Proof.
induction n.
- right. intros (i&?). destruct_fin i.
- decide (P Fin0).
+ left. eauto.
+ specialize (IHn (fun i => P (Fin.FS i))). spec_assert IHn as [IH|IH] by eauto.
* left. destruct IH as (i&IH). exists (Fin.FS i). eauto.
* right. intros (j&H). pose proof (fin_destruct_S j) as [(j'&->) | ->]; eauto.
Qed.
Lemma Frame_withSpace (m n : nat) (sig : Type) (P : SpecV sig n) (P' : SpecV sig m) (I : Vector.t (Fin.t n) m) (ss : Vector.t nat n) (ss' : Vector.t nat m) :
dupfree I ->
Frame (withSpace P ss) I (withSpace P' ss') = withSpace (Frame P I P') (fill I ss ss').
Proof.
intros Hdup. unfold Frame,withSpace.
eapply VectorSpec.eq_nth_iff; intros ? i ->.
simpl_vector.
decide (exists j, I[@j]=i) as [(j&Hj)|Hj].
+ erewrite !fill_correct_nth by eauto. now simpl_vector.
+ assert (not_index I i).
{ hnf. intros (k&<-) % vect_nth_In'. contradict Hj. eauto. }
erewrite !fill_not_index by eauto. now simpl_vector.
Qed.
Lemma tspec_Frame_withSpace'
(m n : nat) (I : Vector.t (Fin.t n) m):
dupfree I -> forall (sig : Type) Q (P : SpecV sig n) (P' : SpecV sig m) (ss : Vector.t nat n) (ss' : Vector.t nat m),
Entails ≃≃( Q , Frame (withSpace P ss) I (withSpace P' ss')) ≃≃( Q, withSpace (Frame P I P') (fill I ss ss')).
Proof. intros H1 **. erewrite <- Frame_withSpace; eauto. Qed.
Lemma LiftTapes_Spec_space (sig F : finType) (m n : nat) (I : Vector.t (Fin.t n) m) P' (P : SpecV sig n) Q' (Q : F -> SpecV sig m) (pM : pTM sig^+ F m)
(ss : Vector.t nat n) (ss' : Vector.t nat m) :
dupfree I ->
Triple (tspec (P',withSpace (Downlift P I) (select I ss))) pM (fun y => tspec (Q' y,withSpace (Q y) ss')) ->
Triple (tspec (P',withSpace P ss)) (LiftTapes pM I) (fun y => tspec (Q' y,withSpace (Frame P I (Q y)) (fill I ss ss'))).
Proof.
intros H1 H2. rewrite <- Downlift_withSpace in H2. apply LiftTapes_Spec in H2. setoid_rewrite tspec_Frame_withSpace' in H2. all:eauto.
Qed.
Lemma LiftTapes_SpecT_space (sig F : finType) (m n : nat) (I : Vector.t (Fin.t n) m) P' (P : SpecV sig n) (k : nat) Q' (Q : F -> SpecV sig m) (pM : pTM sig^+ F m)
(ss : Vector.t nat n) (ss' : Vector.t nat m) :
dupfree I ->
TripleT (tspec (P',withSpace (Downlift P I) (select I ss))) k pM (fun y => tspec (Q' y,withSpace (Q y) ss')) ->
TripleT (tspec (P',withSpace P ss)) k (LiftTapes pM I) (fun y => tspec (Q' y,withSpace (Frame P I (Q y)) (fill I ss ss'))).
Proof.
intros H1 H2. rewrite <- Downlift_withSpace in H2. apply LiftTapes_SpecT in H2. setoid_rewrite tspec_Frame_withSpace' in H2. all:eauto.
Qed.
Lemma LiftTapes_Spec_space_con (sig : finType) (F : finType) (m n : nat) (I : Vector.t (Fin.t n) m)
P' (P : SpecV sig n) Q' (Q : F -> SpecV sig m) R' (R : F -> SpecV sig n) (ss : Vector.t nat n) (ss' : Vector.t nat m) (ss'' : Vector.t nat n)
(pM : pTM sig^+ F m) :
dupfree I ->
Triple (tspec (P',withSpace (Downlift P I) (select I ss))) pM (fun y => tspec (Q' y,withSpace (Q y) ss')) ->
(forall yout, Entails (tspec (Q' yout,withSpace (Frame P I (Q yout)) (fill I ss ss'))) (tspec (R' yout,withSpace (R yout) ss''))) ->
Triple (tspec (P',withSpace P ss)) (LiftTapes pM I) (fun y => tspec (R' y,withSpace (R y) ss'')).
Proof.
intros H1 H2 <-%asPointwise. rewrite <- Downlift_withSpace in H2. apply LiftTapes_Spec in H2.
setoid_rewrite tspec_Frame_withSpace' in H2. all:easy.
Qed.
Lemma LiftTapes_SpecT_space_con (sig : finType) (F : finType) (m n : nat) (I : Vector.t (Fin.t n) m)
P' (P : SpecV sig n) Q' (Q : F -> SpecV sig m) R' (R : F -> SpecV sig n) (ss : Vector.t nat n) (ss' : Vector.t nat m) (ss'' : Vector.t nat n)
(k : nat) (pM : pTM sig^+ F m) :
dupfree I ->
TripleT (tspec (P',withSpace (Downlift P I) (select I ss))) k pM (fun y => tspec (Q' y,withSpace (Q y) ss')) ->
(forall yout, Entails (tspec (Q' yout,withSpace (Frame P I (Q yout)) (fill I ss ss'))) (tspec (R' yout,withSpace (R yout) ss''))) ->
TripleT (tspec (P',withSpace P ss)) k (LiftTapes pM I) (fun y => tspec (R' y,withSpace (R y) ss'')).
Proof.
intros H1 H2 <-%asPointwise. rewrite <- Downlift_withSpace in H2. apply LiftTapes_SpecT in H2.
setoid_rewrite tspec_Frame_withSpace' in H2. all:easy.
Qed.
Section AlphabetLifting.
Variable (sig tau : Type).
Variable (retr : Retract sig tau).
Definition LiftSpec_single (T : RegSpec sig) : RegSpec tau :=
match T with
| Contains r x => Contains (ComposeRetract retr r) x
| Contains_size r x s => Contains_size (ComposeRetract retr r) x s
| Void => Void
| Void_size s => Void_size s
| Custom p => Custom (fun t => p (surjectTape (Retr_g) (inl UNKNOWN) t))
end.
Variable (n : nat).
Definition LiftSpec (T : SpecV sig n) : SpecV tau n :=
Vector.map LiftSpec_single T.
Lemma LiftSpec_surjectTape_tspec_single t T :
tspec_single (LiftSpec_single T) t ->
tspec_single T (surjectTape Retr_g (inl UNKNOWN) t).
Proof. destruct T; cbn in *; intros; simpl_surject; eauto. Qed.
Lemma LiftSpec_surjectTape_tspec_single' t T :
tspec_single T (surjectTape Retr_g (inl UNKNOWN) t) ->
tspec_single (LiftSpec_single T) t.
Proof. destruct T; cbn in *; intros; simpl_surject; eauto. Qed.
Lemma LiftSpec_surjectTapes_tspec tin P' P :
tin ≃≃ (P', LiftSpec P) ->
surjectTapes Retr_g (inl UNKNOWN) tin ≃≃ (P',P).
Proof.
intros (H'&H)%tspecE. eapply tspecI. easy.
intros i; specialize (H i); cbn. unfold LiftSpec in *.
simpl_tape in *. now apply LiftSpec_surjectTape_tspec_single.
Qed.
Lemma LiftSpec_surjectTapes_tspec' tin P P':
surjectTapes Retr_g (inl UNKNOWN) tin ≃≃ (P',P) ->
tin ≃≃ (P',LiftSpec P).
Proof.
intros (H'&H)%tspecE. eapply tspecI. easy. unfold LiftSpec in *.
intros i; specialize (H i); cbn.
simpl_tape in *. now apply LiftSpec_surjectTape_tspec_single'.
Qed.
End AlphabetLifting.
Lemma LiftSpec_withSpace_single (sig tau : Type) (I : Retract sig tau) (P : RegSpec sig) (s : nat) :
LiftSpec_single I (withSpace_single P s) = withSpace_single (LiftSpec_single I P) s.
Proof. destruct P; cbn; eauto. Qed.
Lemma LiftSpec_withSpace (sig tau : Type) (n : nat) (I : Retract sig tau) (P : SpecV sig n) (ss : Vector.t nat n) :
LiftSpec I (withSpace P ss) = withSpace (LiftSpec I P) ss.
Proof.
eapply VectorSpec.eq_nth_iff; intros ? ? ->. unfold LiftSpec, withSpace.
simpl_vector. apply LiftSpec_withSpace_single.
Qed.
Section AlphabetLifting'.
Variable (sig tau : finType) (n : nat).
Variable (retr : Retract sig tau).
Lemma ChangeAlphabet_Spec_ex (F : finType) X P' (Ctx : X -> Prop -> Prop) (P : SpecV sig n) (pM : pTM sig^+ F n) Q' (Q : X -> F -> SpecV sig n) :
Triple (tspec (P',P)) pM (fun y t => exists x:X, Ctx x (tspec (Q' x y,Q x y) t)) ->
(forall x, Proper (Basics.impl ==> Basics.impl) (Ctx x)) ->
Triple (tspec (P',LiftSpec retr P)) (ChangeAlphabet pM retr)
(fun yout t => exists x, Ctx x (tspec (Q' x yout,LiftSpec retr (Q x yout)) t)).
Proof.
rewrite !Triple_iff. intros HTrip HCtx. eapply Realise_monotone.
- TM_Correct. eassumption.
- intros tin (yout, tout) H Henc. cbn in *.
spec_assert H by now apply LiftSpec_surjectTapes_tspec.
destruct H as (x&H). exists x. cbv in HCtx. eapply HCtx. 2:apply H.
now apply LiftSpec_surjectTapes_tspec'.
Qed.
Lemma ChangeAlphabet_Spec (F : finType) P' (P : SpecV sig n) (pM : pTM sig^+ F n) Q' (Q : F -> SpecV sig n) :
Triple (tspec (P',P)) pM (fun yout => tspec (Q' yout,Q yout)) ->
Triple (tspec (P',LiftSpec retr P)) (ChangeAlphabet pM retr) (fun yout => tspec (Q' yout,LiftSpec retr (Q yout))).
Proof.
rewrite !Triple_iff. intros HTrip. eapply Realise_monotone.
- TM_Correct. eassumption.
- intros tin (yout, tout) H Henc. cbn in *.
spec_assert H by now apply LiftSpec_surjectTapes_tspec.
now apply LiftSpec_surjectTapes_tspec'.
Qed.
Lemma ChangeAlphabet_SpecT (F : finType) P' (P : SpecV sig n) (k : nat) (pM : pTM sig^+ F n) Q' (Q : F -> SpecV sig n) :
TripleT (tspec (P',P)) k pM (fun yout => tspec (Q' yout, Q yout)) ->
TripleT (tspec (P',LiftSpec retr P)) k (ChangeAlphabet pM retr) (fun yout => tspec (Q' yout,LiftSpec retr (Q yout))).
Proof.
intros HTrip. split.
{ apply ChangeAlphabet_Spec. eapply TripleT_Triple; eauto. }
{
eapply TerminatesIn_monotone.
- TM_Correct. apply HTrip.
- unfold Triple_TRel. intros tin k' (H&Hk). cbn. split; auto.
now apply LiftSpec_surjectTapes_tspec.
}
Qed.
Lemma ChangeAlphabet_Spec_pre_post (F : finType)
P0 (P : SpecV sig n) (P' : SpecV tau n)
(pM : pTM sig^+ F n)
Q0 (Q : F -> SpecV sig n) (Q' : F -> SpecV tau n) :
Triple (tspec (P0,P)) pM (fun yout => tspec (Q0 yout, Q yout) ) ->
(Entails ≃≃( P0,P') ≃≃( P0,LiftSpec retr P)) ->
(forall yout, Entails ≃≃( Q0 yout, LiftSpec retr (Q yout)) ≃≃( (Q0 yout,Q' yout))) ->
Triple (tspec (P0, P')) (ChangeAlphabet pM retr) (fun yout => tspec (Q0 yout, Q' yout)).
Proof.
intros H1 H2 H3.
eapply Consequence.
- apply ChangeAlphabet_Spec. apply H1.
- apply H2.
- apply H3.
Qed.
Lemma ChangeAlphabet_SpecT_pre_post (F : finType)
P0 (P : SpecV sig n) (P' : SpecV tau n)
(k : nat) (pM : pTM sig^+ F n)
Q0 (Q : F -> SpecV sig n) (Q' : F -> SpecV tau n) :
TripleT (tspec (P0,P)) k pM (fun yout => tspec (Q0 yout, Q yout) ) ->
(Entails ≃≃( P0,P') ≃≃( P0,LiftSpec retr P)) ->
(forall yout, Entails ≃≃( Q0 yout, LiftSpec retr (Q yout)) ≃≃( (Q0 yout,Q' yout))) ->
TripleT (tspec (P0, P')) k (ChangeAlphabet pM retr) (fun yout => tspec (Q0 yout, Q' yout)).
Proof.
intros H1 H2 H3.
eapply ConsequenceT.
- apply ChangeAlphabet_SpecT. apply H1.
- apply H2.
- apply H3.
- reflexivity.
Qed.
Lemma ChangeAlphabet_Spec_pre (F : finType)
P0 (P : SpecV sig n) (P' : SpecV tau n)
(pM : pTM sig^+ F n)
Q0 (Q : F -> SpecV sig n) :
Triple (tspec (P0,P)) pM (fun yout => tspec (Q0 yout, Q yout)) ->
(Entails ≃≃( P0,P') ≃≃( P0, LiftSpec retr P)) ->
Triple (tspec (P0,P')) (ChangeAlphabet pM retr) (fun yout => tspec (Q0 yout,LiftSpec retr (Q yout))).
Proof.
intros H1 H2.
eapply Consequence.
- apply ChangeAlphabet_Spec. apply H1.
- apply H2.
- eauto.
Qed.
Lemma ChangeAlphabet_SpecT_pre (F : finType)
P0 (P : SpecV sig n) (P' : SpecV tau n)
(k : nat) (pM : pTM sig^+ F n)
Q0 (Q : F -> SpecV sig n) :
TripleT (tspec (P0,P)) k pM (fun yout => tspec (Q0 yout, Q yout)) ->
(Entails ≃≃( P0, P') ≃≃( P0,LiftSpec retr P)) ->
TripleT (tspec (P0,P')) k (ChangeAlphabet pM retr) (fun yout => tspec (Q0 yout,LiftSpec retr (Q yout))).
Proof.
intros H1 H2.
eapply ConsequenceT.
- apply ChangeAlphabet_SpecT. apply H1.
- apply H2.
- eauto.
- reflexivity.
Qed.
Lemma ChangeAlphabet_Spec_space_pre_post (F : finType)
P0 (P : SpecV sig n) (P' : SpecV tau n)
(pM : pTM sig^+ F n)
Q0 (Q : F -> SpecV sig n) (Q' : F -> SpecV tau n)
(ss ss' : Vector.t nat n) :
Triple (tspec (P0,withSpace P ss)) pM (fun yout => tspec (Q0 yout,withSpace (Q yout) ss')) ->
(Entails ≃≃( P0, withSpace P' ss) ≃≃( P0, withSpace (LiftSpec retr P) ss)) ->
(forall yout, Entails ≃≃( Q0 yout, withSpace (LiftSpec retr (Q yout)) ss') (tspec (Q0 yout,withSpace (Q' yout) ss'))) ->
Triple (tspec (P0,withSpace P' ss)) (ChangeAlphabet pM retr) (fun yout => tspec (Q0 yout, withSpace (Q' yout) ss')).
Proof.
intros H1 H2 H3.
eapply Consequence.
- apply ChangeAlphabet_Spec. apply H1.
- rewrite LiftSpec_withSpace. apply H2.
- setoid_rewrite Entails_iff. cbn. intros. rewrite LiftSpec_withSpace in H. now apply H3.
Qed.
Lemma ChangeAlphabet_SpecT_space_pre_post (F : finType)
P0 (P : SpecV sig n) (P' : SpecV tau n)
(k : nat) (pM : pTM sig^+ F n)
Q0 (Q : F -> SpecV sig n) (Q' : F -> SpecV tau n)
(ss ss' : Vector.t nat n) :
TripleT (tspec (P0,withSpace P ss)) k pM (fun yout => tspec (Q0 yout,withSpace (Q yout) ss')) ->
(Entails ≃≃( P0, withSpace P' ss) ≃≃( P0, withSpace (LiftSpec retr P) ss)) ->
(forall yout, Entails ≃≃( Q0 yout, withSpace (LiftSpec retr (Q yout)) ss') (tspec (Q0 yout,withSpace (Q' yout) ss'))) ->
TripleT (tspec (P0,withSpace P' ss)) k (ChangeAlphabet pM retr) (fun yout => tspec (Q0 yout, withSpace (Q' yout) ss')).
Proof.
intros H1 H2 H3.
eapply ConsequenceT.
- apply ChangeAlphabet_SpecT. apply H1.
- rewrite LiftSpec_withSpace. apply H2.
- setoid_rewrite Entails_iff. cbn. intros. rewrite LiftSpec_withSpace in H. now apply H3.
- reflexivity.
Qed.
Lemma ChangeAlphabet_Spec_space_pre (F : finType)
P0 (P : SpecV sig n) (P' : SpecV tau n)
(pM : pTM sig^+ F n)
Q0 (Q : F -> SpecV sig n)
(ss ss' : Vector.t nat n) :
Triple (tspec (P0,withSpace P ss)) pM (fun yout => tspec (Q0 yout,withSpace (Q yout) ss')) ->
Entails ≃≃( P0, withSpace P' ss) ≃≃( P0, withSpace (LiftSpec retr P) ss) ->
Triple (tspec (P0, withSpace P' ss)) (ChangeAlphabet pM retr) (fun yout => tspec (Q0 yout, withSpace (LiftSpec retr (Q yout)) ss')).
Proof.
intros H1 H2.
eapply Consequence.
- apply ChangeAlphabet_Spec. apply H1.
- rewrite LiftSpec_withSpace. apply H2.
- setoid_rewrite Entails_iff. cbn. intros. rewrite LiftSpec_withSpace in H. now apply H.
Qed.
Lemma ChangeAlphabet_SpecT_space_pre (F : finType)
P0 (P : SpecV sig n) (P' : SpecV tau n)
(k : nat) (pM : pTM sig^+ F n)
Q0 (Q : F -> SpecV sig n)
(ss ss' : Vector.t nat n) :
TripleT (tspec (P0,withSpace P ss)) k pM (fun yout => tspec (Q0 yout, withSpace (Q yout) ss')) ->
Entails ≃≃( P0, withSpace P' ss) ≃≃( P0, withSpace (LiftSpec retr P) ss) ->
TripleT (tspec (P0, withSpace P' ss)) k (ChangeAlphabet pM retr) (fun yout => tspec (Q0 yout,withSpace (LiftSpec retr (Q yout)) ss')).
Proof.
intros H1 H2.
eapply ConsequenceT.
- apply ChangeAlphabet_SpecT. apply H1.
- rewrite LiftSpec_withSpace. apply H2.
- setoid_rewrite Entails_iff. cbn. intros. rewrite LiftSpec_withSpace in H. now apply H.
- reflexivity.
Qed.
End AlphabetLifting'.
Global Arguments withSpace : simpl never.
Notation "'SpecFalse'" := ([False],_): spec_scope.