Library Paco.paco0_upto
Require Export Program.Basics. Open Scope program_scope.
Require Import paco0.
Set Implicit Arguments.
Section Respectful0.
Local Notation rel := (rel0).
Variable gf: rel -> rel.
Hypothesis gf_mon: monotone0 gf.
Inductive sound0 (clo: rel -> rel): Prop :=
| sound0_intro
(MON: monotone0 clo)
(SOUND:
forall r (PFIX: r <0= gf (clo r)),
r <0= paco0 gf bot0)
.
Hint Constructors sound0.
Structure respectful0 (clo: rel -> rel) : Prop :=
respectful0_intro {
MON: monotone0 clo;
RESPECTFUL:
forall l r (LE: l <0= r) (GF: l <0= gf r),
clo l <0= gf (clo r);
}.
Hint Constructors respectful0.
Inductive gres0 (r: rel) : Prop :=
| gres0_intro
clo
(RES: respectful0 clo)
(CLO: clo r)
.
Hint Constructors gres0.
Lemma gfclo0_mon: forall clo, sound0 clo -> monotone0 (compose gf clo).
Proof.
intros; destruct H; red; intros.
eapply gf_mon; [apply IN|intros; eapply MON0; [apply PR|apply LE]].
Qed.
Hint Resolve gfclo0_mon : paco.
Lemma sound0_is_gf: forall clo (UPTO: sound0 clo),
paco0 (compose gf clo) bot0 <0= paco0 gf bot0.
Proof.
intros. _punfold PR; [|apply gfclo0_mon, UPTO]. edestruct UPTO.
eapply (SOUND (paco0 (compose gf clo) bot0)).
- intros. _punfold PR0; [|apply gfclo0_mon, UPTO].
eapply (gfclo0_mon UPTO); [apply PR0| intros; destruct PR1; [apply H|destruct H]].
- pfold. apply PR.
Qed.
Lemma respectful0_is_sound0: respectful0 <1= sound0.
Proof.
intro clo.
set (rclo := fix rclo clo n (r: rel) :=
match n with
| 0 => r
| S n' => rclo clo n' r \0/ clo (rclo clo n' r)
end).
intros. destruct PR. econstructor; [apply MON0|].
intros. set (rr := exists n, rclo clo n r).
assert (rr) by (exists 0; apply PR); clear PR.
cut (forall n, rclo clo n r <0= gf (rclo clo (S n) r)).
{ intro X; revert H; pcofix CIH; intros.
unfold rr in *; destruct H0.
pfold. eapply gf_mon.
- apply X. apply H.
- intros. right. apply CIH. exists (S x). apply PR.
}
induction n; intros.
- eapply gf_mon.
+ clear RESPECTFUL0. eapply PFIX, PR.
+ intros. right. eapply PR0.
- destruct PR.
+ eapply gf_mon; [eapply IHn, H0|]. intros. left. apply PR.
+ eapply gf_mon; [eapply RESPECTFUL0; [|apply IHn|]|]; intros.
* left; apply PR.
* apply H0.
* right; apply PR.
Qed.
Lemma respectful0_compose
clo0 clo1
(RES0: respectful0 clo0)
(RES1: respectful0 clo1):
respectful0 (compose clo0 clo1).
Proof.
intros. destruct RES0, RES1.
econstructor.
- repeat intro. eapply MON0; [apply IN|].
intros. eapply MON1; [apply PR|apply LE].
- intros. eapply RESPECTFUL0; [| |apply PR].
+ intros. eapply MON1; [apply PR0|apply LE].
+ intros. eapply RESPECTFUL1; [apply LE| apply GF| apply PR0].
Qed.
Lemma grespectful0_mon: monotone0 gres0.
Proof.
red. intros.
destruct IN; destruct RES; exists clo; [|eapply MON0; [eapply CLO| eapply LE]].
constructor; [eapply MON0|].
intros. eapply RESPECTFUL0; [apply LE0|apply GF|apply PR].
Qed.
Lemma grespectful0_respectful0: respectful0 gres0.
Proof.
econstructor; [apply grespectful0_mon|intros].
destruct PR; destruct RES; eapply gf_mon with (r:=clo r).
eapply RESPECTFUL0; [apply LE|apply GF|apply CLO].
intros. econstructor; [constructor; [apply MON0|apply RESPECTFUL0]|apply PR].
Qed.
Lemma gfgres0_mon: monotone0 (compose gf gres0).
Proof.
destruct grespectful0_respectful0.
unfold monotone0. intros. eapply gf_mon; [eapply IN|].
intros. eapply MON0;[apply PR|apply LE].
Qed.
Hint Resolve gfgres0_mon : paco.
Lemma grespectful0_greatest: forall clo (RES: respectful0 clo), clo <1= gres0.
Proof. intros. econstructor;[apply RES|apply PR]. Qed.
Lemma grespectful0_incl: forall r, r <0= gres0 r.
Proof.
intros; eexists (fun x => x).
- econstructor.
+ red; intros; apply LE, IN.
+ intros; apply GF, PR0.
- apply PR.
Qed.
Hint Resolve grespectful0_incl.
Lemma grespectful0_compose: forall clo (RES: respectful0 clo) r,
clo (gres0 r) <0= gres0 r.
Proof.
intros; eapply grespectful0_greatest with (clo := compose clo gres0); [|apply PR].
apply respectful0_compose; [apply RES|apply grespectful0_respectful0].
Qed.
Lemma grespectful0_incl_rev: forall r,
gres0 (paco0 (compose gf gres0) r) <0= paco0 (compose gf gres0) r.
Proof.
intro r; pcofix CIH; intros; pfold.
eapply gf_mon, grespectful0_compose, grespectful0_respectful0.
destruct grespectful0_respectful0; eapply RESPECTFUL0, PR; intros; [apply grespectful0_incl; right; apply CIH, grespectful0_incl, PR0|].
_punfold PR0; [|apply gfgres0_mon].
eapply gfgres0_mon; [apply PR0|].
intros; destruct PR1.
- left. eapply paco0_mon; [apply H| apply CIH0].
- right. eapply CIH0, H.
Qed.
Inductive rclo0 clo (r: rel): rel :=
| rclo0_incl
(R: r):
@rclo0 clo r
| rclo0_step'
r'
(R': r' <0= rclo0 clo r)
(CLOR':clo r'):
@rclo0 clo r
| rclo0_gf
r'
(R': r' <0= rclo0 clo r)
(CLOR':@gf r'):
@rclo0 clo r
.
Lemma rclo0_mon clo:
monotone0 (rclo0 clo).
Proof.
repeat intro. induction IN.
- econstructor 1. apply LE, R.
- econstructor 2; [intros; eapply H, PR| eapply CLOR'].
- econstructor 3; [intros; eapply H, PR| eapply CLOR'].
Qed.
Hint Resolve rclo0_mon: paco.
Lemma rclo0_base
clo
(MON: monotone0 clo):
clo <1= rclo0 clo.
Proof.
intros. econstructor 2; [intros; apply PR0|].
eapply MON; [apply PR|intros; constructor; apply PR0].
Qed.
Lemma rclo0_step
(clo: rel -> rel) r:
clo (rclo0 clo r) <0= rclo0 clo r.
Proof.
intros. econstructor 2; [intros; apply PR0|apply PR].
Qed.
Lemma rclo0_rclo0
clo r
(MON: monotone0 clo):
rclo0 clo (rclo0 clo r) <0= rclo0 clo r.
Proof.
intros. induction PR.
- eapply R.
- econstructor 2; [eapply H | eapply CLOR'].
- econstructor 3; [eapply H | eapply CLOR'].
Qed.
Structure weak_respectful0 (clo: rel -> rel) : Prop :=
weak_respectful0_intro {
WEAK_MON: monotone0 clo;
WEAK_RESPECTFUL:
forall l r (LE: l <0= r) (GF: l <0= gf r),
clo l <0= gf (rclo0 clo r);
}.
Hint Constructors weak_respectful0.
Lemma weak_respectful0_respectful0
clo (RES: weak_respectful0 clo):
respectful0 (rclo0 clo).
Proof.
inversion RES. econstructor; [eapply rclo0_mon|]. intros.
induction PR; intros.
- eapply gf_mon; [apply GF, R|]. intros.
apply rclo0_incl. apply PR.
- eapply gf_mon.
+ eapply WEAK_RESPECTFUL0; [|apply H|apply CLOR'].
intros. eapply rclo0_mon; [apply R', PR|apply LE].
+ intros. apply rclo0_rclo0;[apply WEAK_MON0|apply PR].
- eapply gf_mon; [apply CLOR'|].
intros. eapply rclo0_mon; [apply R', PR| apply LE].
Qed.
Lemma upto0_init:
paco0 (compose gf gres0) bot0 <0= paco0 gf bot0.
Proof.
apply sound0_is_gf.
apply respectful0_is_sound0.
apply grespectful0_respectful0.
Qed.
Lemma upto0_final:
paco0 gf <1= paco0 (compose gf gres0).
Proof.
pcofix CIH. intros. _punfold PR; [|apply gf_mon]. pfold.
eapply gf_mon; [|apply grespectful0_incl].
eapply gf_mon; [apply PR|]. intros. right.
inversion PR0; [apply CIH, H | apply CIH0, H].
Qed.
Lemma upto0_step
r clo (RES: weak_respectful0 clo):
clo (paco0 (compose gf gres0) r) <0= paco0 (compose gf gres0) r.
Proof.
intros. apply grespectful0_incl_rev.
assert (RES' := weak_respectful0_respectful0 RES).
eapply grespectful0_greatest; [apply RES'|].
eapply rclo0_base; [apply RES|].
inversion RES. apply PR.
Qed.
Lemma upto0_step_under
r clo (RES: weak_respectful0 clo):
clo (gres0 r) <0= gres0 r.
Proof.
intros. apply weak_respectful0_respectful0 in RES.
eapply grespectful0_compose; [apply RES|].
econstructor 2; [intros; constructor 1; apply PR0 | apply PR].
Qed.
End Respectful0.
Lemma grespectful0_impl (gf gf': rel0 -> rel0) r
(PR: gres0 gf r)
(EQ: forall r, gf r <-> gf' r):
gres0 gf' r.
Proof.
intros. destruct PR. econstructor; [|apply CLO].
destruct RES. econstructor; [apply MON0|].
intros. eapply EQ. eapply RESPECTFUL0; [apply LE| |apply PR].
intros. eapply EQ. apply GF, PR0.
Qed.
Lemma grespectful0_iff (gf gf': rel0 -> rel0) r
(EQ: forall r, gf r <-> gf' r):
gres0 gf r <-> gres0 gf' r.
Proof.
split; intros.
- eapply grespectful0_impl; [apply H | apply EQ].
- eapply grespectful0_impl; [apply H | split; apply EQ].
Qed.
Hint Constructors sound0.
Hint Constructors respectful0.
Hint Constructors gres0.
Hint Resolve gfclo0_mon : paco.
Hint Resolve gfgres0_mon : paco.
Hint Resolve grespectful0_incl.
Hint Resolve rclo0_mon: paco.
Hint Constructors weak_respectful0.
Ltac pupto0_init := eapply upto0_init; [eauto with paco|].
Ltac pupto0_final := first [eapply upto0_final; [eauto with paco|] | eapply grespectful0_incl].
Ltac pupto0 H := first [eapply upto0_step|eapply upto0_step_under]; [|eapply H|]; [eauto with paco|].
Require Import paco0.
Set Implicit Arguments.
Section Respectful0.
Local Notation rel := (rel0).
Variable gf: rel -> rel.
Hypothesis gf_mon: monotone0 gf.
Inductive sound0 (clo: rel -> rel): Prop :=
| sound0_intro
(MON: monotone0 clo)
(SOUND:
forall r (PFIX: r <0= gf (clo r)),
r <0= paco0 gf bot0)
.
Hint Constructors sound0.
Structure respectful0 (clo: rel -> rel) : Prop :=
respectful0_intro {
MON: monotone0 clo;
RESPECTFUL:
forall l r (LE: l <0= r) (GF: l <0= gf r),
clo l <0= gf (clo r);
}.
Hint Constructors respectful0.
Inductive gres0 (r: rel) : Prop :=
| gres0_intro
clo
(RES: respectful0 clo)
(CLO: clo r)
.
Hint Constructors gres0.
Lemma gfclo0_mon: forall clo, sound0 clo -> monotone0 (compose gf clo).
Proof.
intros; destruct H; red; intros.
eapply gf_mon; [apply IN|intros; eapply MON0; [apply PR|apply LE]].
Qed.
Hint Resolve gfclo0_mon : paco.
Lemma sound0_is_gf: forall clo (UPTO: sound0 clo),
paco0 (compose gf clo) bot0 <0= paco0 gf bot0.
Proof.
intros. _punfold PR; [|apply gfclo0_mon, UPTO]. edestruct UPTO.
eapply (SOUND (paco0 (compose gf clo) bot0)).
- intros. _punfold PR0; [|apply gfclo0_mon, UPTO].
eapply (gfclo0_mon UPTO); [apply PR0| intros; destruct PR1; [apply H|destruct H]].
- pfold. apply PR.
Qed.
Lemma respectful0_is_sound0: respectful0 <1= sound0.
Proof.
intro clo.
set (rclo := fix rclo clo n (r: rel) :=
match n with
| 0 => r
| S n' => rclo clo n' r \0/ clo (rclo clo n' r)
end).
intros. destruct PR. econstructor; [apply MON0|].
intros. set (rr := exists n, rclo clo n r).
assert (rr) by (exists 0; apply PR); clear PR.
cut (forall n, rclo clo n r <0= gf (rclo clo (S n) r)).
{ intro X; revert H; pcofix CIH; intros.
unfold rr in *; destruct H0.
pfold. eapply gf_mon.
- apply X. apply H.
- intros. right. apply CIH. exists (S x). apply PR.
}
induction n; intros.
- eapply gf_mon.
+ clear RESPECTFUL0. eapply PFIX, PR.
+ intros. right. eapply PR0.
- destruct PR.
+ eapply gf_mon; [eapply IHn, H0|]. intros. left. apply PR.
+ eapply gf_mon; [eapply RESPECTFUL0; [|apply IHn|]|]; intros.
* left; apply PR.
* apply H0.
* right; apply PR.
Qed.
Lemma respectful0_compose
clo0 clo1
(RES0: respectful0 clo0)
(RES1: respectful0 clo1):
respectful0 (compose clo0 clo1).
Proof.
intros. destruct RES0, RES1.
econstructor.
- repeat intro. eapply MON0; [apply IN|].
intros. eapply MON1; [apply PR|apply LE].
- intros. eapply RESPECTFUL0; [| |apply PR].
+ intros. eapply MON1; [apply PR0|apply LE].
+ intros. eapply RESPECTFUL1; [apply LE| apply GF| apply PR0].
Qed.
Lemma grespectful0_mon: monotone0 gres0.
Proof.
red. intros.
destruct IN; destruct RES; exists clo; [|eapply MON0; [eapply CLO| eapply LE]].
constructor; [eapply MON0|].
intros. eapply RESPECTFUL0; [apply LE0|apply GF|apply PR].
Qed.
Lemma grespectful0_respectful0: respectful0 gres0.
Proof.
econstructor; [apply grespectful0_mon|intros].
destruct PR; destruct RES; eapply gf_mon with (r:=clo r).
eapply RESPECTFUL0; [apply LE|apply GF|apply CLO].
intros. econstructor; [constructor; [apply MON0|apply RESPECTFUL0]|apply PR].
Qed.
Lemma gfgres0_mon: monotone0 (compose gf gres0).
Proof.
destruct grespectful0_respectful0.
unfold monotone0. intros. eapply gf_mon; [eapply IN|].
intros. eapply MON0;[apply PR|apply LE].
Qed.
Hint Resolve gfgres0_mon : paco.
Lemma grespectful0_greatest: forall clo (RES: respectful0 clo), clo <1= gres0.
Proof. intros. econstructor;[apply RES|apply PR]. Qed.
Lemma grespectful0_incl: forall r, r <0= gres0 r.
Proof.
intros; eexists (fun x => x).
- econstructor.
+ red; intros; apply LE, IN.
+ intros; apply GF, PR0.
- apply PR.
Qed.
Hint Resolve grespectful0_incl.
Lemma grespectful0_compose: forall clo (RES: respectful0 clo) r,
clo (gres0 r) <0= gres0 r.
Proof.
intros; eapply grespectful0_greatest with (clo := compose clo gres0); [|apply PR].
apply respectful0_compose; [apply RES|apply grespectful0_respectful0].
Qed.
Lemma grespectful0_incl_rev: forall r,
gres0 (paco0 (compose gf gres0) r) <0= paco0 (compose gf gres0) r.
Proof.
intro r; pcofix CIH; intros; pfold.
eapply gf_mon, grespectful0_compose, grespectful0_respectful0.
destruct grespectful0_respectful0; eapply RESPECTFUL0, PR; intros; [apply grespectful0_incl; right; apply CIH, grespectful0_incl, PR0|].
_punfold PR0; [|apply gfgres0_mon].
eapply gfgres0_mon; [apply PR0|].
intros; destruct PR1.
- left. eapply paco0_mon; [apply H| apply CIH0].
- right. eapply CIH0, H.
Qed.
Inductive rclo0 clo (r: rel): rel :=
| rclo0_incl
(R: r):
@rclo0 clo r
| rclo0_step'
r'
(R': r' <0= rclo0 clo r)
(CLOR':clo r'):
@rclo0 clo r
| rclo0_gf
r'
(R': r' <0= rclo0 clo r)
(CLOR':@gf r'):
@rclo0 clo r
.
Lemma rclo0_mon clo:
monotone0 (rclo0 clo).
Proof.
repeat intro. induction IN.
- econstructor 1. apply LE, R.
- econstructor 2; [intros; eapply H, PR| eapply CLOR'].
- econstructor 3; [intros; eapply H, PR| eapply CLOR'].
Qed.
Hint Resolve rclo0_mon: paco.
Lemma rclo0_base
clo
(MON: monotone0 clo):
clo <1= rclo0 clo.
Proof.
intros. econstructor 2; [intros; apply PR0|].
eapply MON; [apply PR|intros; constructor; apply PR0].
Qed.
Lemma rclo0_step
(clo: rel -> rel) r:
clo (rclo0 clo r) <0= rclo0 clo r.
Proof.
intros. econstructor 2; [intros; apply PR0|apply PR].
Qed.
Lemma rclo0_rclo0
clo r
(MON: monotone0 clo):
rclo0 clo (rclo0 clo r) <0= rclo0 clo r.
Proof.
intros. induction PR.
- eapply R.
- econstructor 2; [eapply H | eapply CLOR'].
- econstructor 3; [eapply H | eapply CLOR'].
Qed.
Structure weak_respectful0 (clo: rel -> rel) : Prop :=
weak_respectful0_intro {
WEAK_MON: monotone0 clo;
WEAK_RESPECTFUL:
forall l r (LE: l <0= r) (GF: l <0= gf r),
clo l <0= gf (rclo0 clo r);
}.
Hint Constructors weak_respectful0.
Lemma weak_respectful0_respectful0
clo (RES: weak_respectful0 clo):
respectful0 (rclo0 clo).
Proof.
inversion RES. econstructor; [eapply rclo0_mon|]. intros.
induction PR; intros.
- eapply gf_mon; [apply GF, R|]. intros.
apply rclo0_incl. apply PR.
- eapply gf_mon.
+ eapply WEAK_RESPECTFUL0; [|apply H|apply CLOR'].
intros. eapply rclo0_mon; [apply R', PR|apply LE].
+ intros. apply rclo0_rclo0;[apply WEAK_MON0|apply PR].
- eapply gf_mon; [apply CLOR'|].
intros. eapply rclo0_mon; [apply R', PR| apply LE].
Qed.
Lemma upto0_init:
paco0 (compose gf gres0) bot0 <0= paco0 gf bot0.
Proof.
apply sound0_is_gf.
apply respectful0_is_sound0.
apply grespectful0_respectful0.
Qed.
Lemma upto0_final:
paco0 gf <1= paco0 (compose gf gres0).
Proof.
pcofix CIH. intros. _punfold PR; [|apply gf_mon]. pfold.
eapply gf_mon; [|apply grespectful0_incl].
eapply gf_mon; [apply PR|]. intros. right.
inversion PR0; [apply CIH, H | apply CIH0, H].
Qed.
Lemma upto0_step
r clo (RES: weak_respectful0 clo):
clo (paco0 (compose gf gres0) r) <0= paco0 (compose gf gres0) r.
Proof.
intros. apply grespectful0_incl_rev.
assert (RES' := weak_respectful0_respectful0 RES).
eapply grespectful0_greatest; [apply RES'|].
eapply rclo0_base; [apply RES|].
inversion RES. apply PR.
Qed.
Lemma upto0_step_under
r clo (RES: weak_respectful0 clo):
clo (gres0 r) <0= gres0 r.
Proof.
intros. apply weak_respectful0_respectful0 in RES.
eapply grespectful0_compose; [apply RES|].
econstructor 2; [intros; constructor 1; apply PR0 | apply PR].
Qed.
End Respectful0.
Lemma grespectful0_impl (gf gf': rel0 -> rel0) r
(PR: gres0 gf r)
(EQ: forall r, gf r <-> gf' r):
gres0 gf' r.
Proof.
intros. destruct PR. econstructor; [|apply CLO].
destruct RES. econstructor; [apply MON0|].
intros. eapply EQ. eapply RESPECTFUL0; [apply LE| |apply PR].
intros. eapply EQ. apply GF, PR0.
Qed.
Lemma grespectful0_iff (gf gf': rel0 -> rel0) r
(EQ: forall r, gf r <-> gf' r):
gres0 gf r <-> gres0 gf' r.
Proof.
split; intros.
- eapply grespectful0_impl; [apply H | apply EQ].
- eapply grespectful0_impl; [apply H | split; apply EQ].
Qed.
Hint Constructors sound0.
Hint Constructors respectful0.
Hint Constructors gres0.
Hint Resolve gfclo0_mon : paco.
Hint Resolve gfgres0_mon : paco.
Hint Resolve grespectful0_incl.
Hint Resolve rclo0_mon: paco.
Hint Constructors weak_respectful0.
Ltac pupto0_init := eapply upto0_init; [eauto with paco|].
Ltac pupto0_final := first [eapply upto0_final; [eauto with paco|] | eapply grespectful0_incl].
Ltac pupto0 H := first [eapply upto0_step|eapply upto0_step_under]; [|eapply H|]; [eauto with paco|].