Library Paco.paco2_upto
Require Export Program.Basics. Open Scope program_scope.
Require Import paco2.
Set Implicit Arguments.
Section Respectful2.
Variable T0 : Type.
Variable T1 : forall (x0: @T0), Type.
Local Notation rel := (rel2 T0 T1).
Variable gf: rel -> rel.
Hypothesis gf_mon: monotone2 gf.
Inductive sound2 (clo: rel -> rel): Prop :=
| sound2_intro
(MON: monotone2 clo)
(SOUND:
forall r (PFIX: r <2= gf (clo r)),
r <2= paco2 gf bot2)
.
Hint Constructors sound2.
Structure respectful2 (clo: rel -> rel) : Prop :=
respectful2_intro {
MON: monotone2 clo;
RESPECTFUL:
forall l r (LE: l <2= r) (GF: l <2= gf r),
clo l <2= gf (clo r);
}.
Hint Constructors respectful2.
Inductive gres2 (r: rel) e0 e1 : Prop :=
| gres2_intro
clo
(RES: respectful2 clo)
(CLO: clo r e0 e1)
.
Hint Constructors gres2.
Lemma gfclo2_mon: forall clo, sound2 clo -> monotone2 (compose gf clo).
Proof.
intros; destruct H; red; intros.
eapply gf_mon; [apply IN|intros; eapply MON0; [apply PR|apply LE]].
Qed.
Hint Resolve gfclo2_mon : paco.
Lemma sound2_is_gf: forall clo (UPTO: sound2 clo),
paco2 (compose gf clo) bot2 <2= paco2 gf bot2.
Proof.
intros. _punfold PR; [|apply gfclo2_mon, UPTO]. edestruct UPTO.
eapply (SOUND (paco2 (compose gf clo) bot2)).
- intros. _punfold PR0; [|apply gfclo2_mon, UPTO].
eapply (gfclo2_mon UPTO); [apply PR0| intros; destruct PR1; [apply H|destruct H]].
- pfold. apply PR.
Qed.
Lemma respectful2_is_sound2: respectful2 <1= sound2.
Proof.
intro clo.
set (rclo := fix rclo clo n (r: rel) :=
match n with
| 0 => r
| S n' => rclo clo n' r \2/ clo (rclo clo n' r)
end).
intros. destruct PR. econstructor; [apply MON0|].
intros. set (rr e0 e1 := exists n, rclo clo n r e0 e1).
assert (rr x0 x1) by (exists 0; apply PR); clear PR.
cut (forall n, rclo clo n r <2= gf (rclo clo (S n) r)).
{ intro X; revert x0 x1 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 respectful2_compose
clo0 clo1
(RES0: respectful2 clo0)
(RES1: respectful2 clo1):
respectful2 (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 grespectful2_mon: monotone2 gres2.
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 grespectful2_respectful2: respectful2 gres2.
Proof.
econstructor; [apply grespectful2_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 gfgres2_mon: monotone2 (compose gf gres2).
Proof.
destruct grespectful2_respectful2.
unfold monotone2. intros. eapply gf_mon; [eapply IN|].
intros. eapply MON0;[apply PR|apply LE].
Qed.
Hint Resolve gfgres2_mon : paco.
Lemma grespectful2_greatest: forall clo (RES: respectful2 clo), clo <3= gres2.
Proof. intros. econstructor;[apply RES|apply PR]. Qed.
Lemma grespectful2_incl: forall r, r <2= gres2 r.
Proof.
intros; eexists (fun x => x).
- econstructor.
+ red; intros; apply LE, IN.
+ intros; apply GF, PR0.
- apply PR.
Qed.
Hint Resolve grespectful2_incl.
Lemma grespectful2_compose: forall clo (RES: respectful2 clo) r,
clo (gres2 r) <2= gres2 r.
Proof.
intros; eapply grespectful2_greatest with (clo := compose clo gres2); [|apply PR].
apply respectful2_compose; [apply RES|apply grespectful2_respectful2].
Qed.
Lemma grespectful2_incl_rev: forall r,
gres2 (paco2 (compose gf gres2) r) <2= paco2 (compose gf gres2) r.
Proof.
intro r; pcofix CIH; intros; pfold.
eapply gf_mon, grespectful2_compose, grespectful2_respectful2.
destruct grespectful2_respectful2; eapply RESPECTFUL0, PR; intros; [apply grespectful2_incl; right; apply CIH, grespectful2_incl, PR0|].
_punfold PR0; [|apply gfgres2_mon].
eapply gfgres2_mon; [apply PR0|].
intros; destruct PR1.
- left. eapply paco2_mon; [apply H| apply CIH0].
- right. eapply CIH0, H.
Qed.
Inductive rclo2 clo (r: rel): rel :=
| rclo2_incl
e0 e1
(R: r e0 e1):
@rclo2 clo r e0 e1
| rclo2_step'
r' e0 e1
(R': r' <2= rclo2 clo r)
(CLOR':clo r' e0 e1):
@rclo2 clo r e0 e1
| rclo2_gf
r' e0 e1
(R': r' <2= rclo2 clo r)
(CLOR':@gf r' e0 e1):
@rclo2 clo r e0 e1
.
Lemma rclo2_mon clo:
monotone2 (rclo2 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 rclo2_mon: paco.
Lemma rclo2_base
clo
(MON: monotone2 clo):
clo <3= rclo2 clo.
Proof.
intros. econstructor 2; [intros; apply PR0|].
eapply MON; [apply PR|intros; constructor; apply PR0].
Qed.
Lemma rclo2_step
(clo: rel -> rel) r:
clo (rclo2 clo r) <2= rclo2 clo r.
Proof.
intros. econstructor 2; [intros; apply PR0|apply PR].
Qed.
Lemma rclo2_rclo2
clo r
(MON: monotone2 clo):
rclo2 clo (rclo2 clo r) <2= rclo2 clo r.
Proof.
intros. induction PR.
- eapply R.
- econstructor 2; [eapply H | eapply CLOR'].
- econstructor 3; [eapply H | eapply CLOR'].
Qed.
Structure weak_respectful2 (clo: rel -> rel) : Prop :=
weak_respectful2_intro {
WEAK_MON: monotone2 clo;
WEAK_RESPECTFUL:
forall l r (LE: l <2= r) (GF: l <2= gf r),
clo l <2= gf (rclo2 clo r);
}.
Hint Constructors weak_respectful2.
Lemma weak_respectful2_respectful2
clo (RES: weak_respectful2 clo):
respectful2 (rclo2 clo).
Proof.
inversion RES. econstructor; [eapply rclo2_mon|]. intros.
induction PR; intros.
- eapply gf_mon; [apply GF, R|]. intros.
apply rclo2_incl. apply PR.
- eapply gf_mon.
+ eapply WEAK_RESPECTFUL0; [|apply H|apply CLOR'].
intros. eapply rclo2_mon; [apply R', PR|apply LE].
+ intros. apply rclo2_rclo2;[apply WEAK_MON0|apply PR].
- eapply gf_mon; [apply CLOR'|].
intros. eapply rclo2_mon; [apply R', PR| apply LE].
Qed.
Lemma upto2_init:
paco2 (compose gf gres2) bot2 <2= paco2 gf bot2.
Proof.
apply sound2_is_gf.
apply respectful2_is_sound2.
apply grespectful2_respectful2.
Qed.
Lemma upto2_final:
paco2 gf <3= paco2 (compose gf gres2).
Proof.
pcofix CIH. intros. _punfold PR; [|apply gf_mon]. pfold.
eapply gf_mon; [|apply grespectful2_incl].
eapply gf_mon; [apply PR|]. intros. right.
inversion PR0; [apply CIH, H | apply CIH0, H].
Qed.
Lemma upto2_step
r clo (RES: weak_respectful2 clo):
clo (paco2 (compose gf gres2) r) <2= paco2 (compose gf gres2) r.
Proof.
intros. apply grespectful2_incl_rev.
assert (RES' := weak_respectful2_respectful2 RES).
eapply grespectful2_greatest; [apply RES'|].
eapply rclo2_base; [apply RES|].
inversion RES. apply PR.
Qed.
Lemma upto2_step_under
r clo (RES: weak_respectful2 clo):
clo (gres2 r) <2= gres2 r.
Proof.
intros. apply weak_respectful2_respectful2 in RES.
eapply grespectful2_compose; [apply RES|].
econstructor 2; [intros; constructor 1; apply PR0 | apply PR].
Qed.
End Respectful2.
Lemma grespectful2_impl T0 T1 (gf gf': rel2 T0 T1 -> rel2 T0 T1) r x0 x1
(PR: gres2 gf r x0 x1)
(EQ: forall r x0 x1, gf r x0 x1 <-> gf' r x0 x1):
gres2 gf' r x0 x1.
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 grespectful2_iff T0 T1 (gf gf': rel2 T0 T1 -> rel2 T0 T1) r x0 x1
(EQ: forall r x0 x1, gf r x0 x1 <-> gf' r x0 x1):
gres2 gf r x0 x1 <-> gres2 gf' r x0 x1.
Proof.
split; intros.
- eapply grespectful2_impl; [apply H | apply EQ].
- eapply grespectful2_impl; [apply H | split; apply EQ].
Qed.
Hint Constructors sound2.
Hint Constructors respectful2.
Hint Constructors gres2.
Hint Resolve gfclo2_mon : paco.
Hint Resolve gfgres2_mon : paco.
Hint Resolve grespectful2_incl.
Hint Resolve rclo2_mon: paco.
Hint Constructors weak_respectful2.
Ltac pupto2_init := eapply upto2_init; [eauto with paco|].
Ltac pupto2_final := first [eapply upto2_final; [eauto with paco|] | eapply grespectful2_incl].
Ltac pupto2 H := first [eapply upto2_step|eapply upto2_step_under]; [|eapply H|]; [eauto with paco|].
Require Import paco2.
Set Implicit Arguments.
Section Respectful2.
Variable T0 : Type.
Variable T1 : forall (x0: @T0), Type.
Local Notation rel := (rel2 T0 T1).
Variable gf: rel -> rel.
Hypothesis gf_mon: monotone2 gf.
Inductive sound2 (clo: rel -> rel): Prop :=
| sound2_intro
(MON: monotone2 clo)
(SOUND:
forall r (PFIX: r <2= gf (clo r)),
r <2= paco2 gf bot2)
.
Hint Constructors sound2.
Structure respectful2 (clo: rel -> rel) : Prop :=
respectful2_intro {
MON: monotone2 clo;
RESPECTFUL:
forall l r (LE: l <2= r) (GF: l <2= gf r),
clo l <2= gf (clo r);
}.
Hint Constructors respectful2.
Inductive gres2 (r: rel) e0 e1 : Prop :=
| gres2_intro
clo
(RES: respectful2 clo)
(CLO: clo r e0 e1)
.
Hint Constructors gres2.
Lemma gfclo2_mon: forall clo, sound2 clo -> monotone2 (compose gf clo).
Proof.
intros; destruct H; red; intros.
eapply gf_mon; [apply IN|intros; eapply MON0; [apply PR|apply LE]].
Qed.
Hint Resolve gfclo2_mon : paco.
Lemma sound2_is_gf: forall clo (UPTO: sound2 clo),
paco2 (compose gf clo) bot2 <2= paco2 gf bot2.
Proof.
intros. _punfold PR; [|apply gfclo2_mon, UPTO]. edestruct UPTO.
eapply (SOUND (paco2 (compose gf clo) bot2)).
- intros. _punfold PR0; [|apply gfclo2_mon, UPTO].
eapply (gfclo2_mon UPTO); [apply PR0| intros; destruct PR1; [apply H|destruct H]].
- pfold. apply PR.
Qed.
Lemma respectful2_is_sound2: respectful2 <1= sound2.
Proof.
intro clo.
set (rclo := fix rclo clo n (r: rel) :=
match n with
| 0 => r
| S n' => rclo clo n' r \2/ clo (rclo clo n' r)
end).
intros. destruct PR. econstructor; [apply MON0|].
intros. set (rr e0 e1 := exists n, rclo clo n r e0 e1).
assert (rr x0 x1) by (exists 0; apply PR); clear PR.
cut (forall n, rclo clo n r <2= gf (rclo clo (S n) r)).
{ intro X; revert x0 x1 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 respectful2_compose
clo0 clo1
(RES0: respectful2 clo0)
(RES1: respectful2 clo1):
respectful2 (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 grespectful2_mon: monotone2 gres2.
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 grespectful2_respectful2: respectful2 gres2.
Proof.
econstructor; [apply grespectful2_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 gfgres2_mon: monotone2 (compose gf gres2).
Proof.
destruct grespectful2_respectful2.
unfold monotone2. intros. eapply gf_mon; [eapply IN|].
intros. eapply MON0;[apply PR|apply LE].
Qed.
Hint Resolve gfgres2_mon : paco.
Lemma grespectful2_greatest: forall clo (RES: respectful2 clo), clo <3= gres2.
Proof. intros. econstructor;[apply RES|apply PR]. Qed.
Lemma grespectful2_incl: forall r, r <2= gres2 r.
Proof.
intros; eexists (fun x => x).
- econstructor.
+ red; intros; apply LE, IN.
+ intros; apply GF, PR0.
- apply PR.
Qed.
Hint Resolve grespectful2_incl.
Lemma grespectful2_compose: forall clo (RES: respectful2 clo) r,
clo (gres2 r) <2= gres2 r.
Proof.
intros; eapply grespectful2_greatest with (clo := compose clo gres2); [|apply PR].
apply respectful2_compose; [apply RES|apply grespectful2_respectful2].
Qed.
Lemma grespectful2_incl_rev: forall r,
gres2 (paco2 (compose gf gres2) r) <2= paco2 (compose gf gres2) r.
Proof.
intro r; pcofix CIH; intros; pfold.
eapply gf_mon, grespectful2_compose, grespectful2_respectful2.
destruct grespectful2_respectful2; eapply RESPECTFUL0, PR; intros; [apply grespectful2_incl; right; apply CIH, grespectful2_incl, PR0|].
_punfold PR0; [|apply gfgres2_mon].
eapply gfgres2_mon; [apply PR0|].
intros; destruct PR1.
- left. eapply paco2_mon; [apply H| apply CIH0].
- right. eapply CIH0, H.
Qed.
Inductive rclo2 clo (r: rel): rel :=
| rclo2_incl
e0 e1
(R: r e0 e1):
@rclo2 clo r e0 e1
| rclo2_step'
r' e0 e1
(R': r' <2= rclo2 clo r)
(CLOR':clo r' e0 e1):
@rclo2 clo r e0 e1
| rclo2_gf
r' e0 e1
(R': r' <2= rclo2 clo r)
(CLOR':@gf r' e0 e1):
@rclo2 clo r e0 e1
.
Lemma rclo2_mon clo:
monotone2 (rclo2 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 rclo2_mon: paco.
Lemma rclo2_base
clo
(MON: monotone2 clo):
clo <3= rclo2 clo.
Proof.
intros. econstructor 2; [intros; apply PR0|].
eapply MON; [apply PR|intros; constructor; apply PR0].
Qed.
Lemma rclo2_step
(clo: rel -> rel) r:
clo (rclo2 clo r) <2= rclo2 clo r.
Proof.
intros. econstructor 2; [intros; apply PR0|apply PR].
Qed.
Lemma rclo2_rclo2
clo r
(MON: monotone2 clo):
rclo2 clo (rclo2 clo r) <2= rclo2 clo r.
Proof.
intros. induction PR.
- eapply R.
- econstructor 2; [eapply H | eapply CLOR'].
- econstructor 3; [eapply H | eapply CLOR'].
Qed.
Structure weak_respectful2 (clo: rel -> rel) : Prop :=
weak_respectful2_intro {
WEAK_MON: monotone2 clo;
WEAK_RESPECTFUL:
forall l r (LE: l <2= r) (GF: l <2= gf r),
clo l <2= gf (rclo2 clo r);
}.
Hint Constructors weak_respectful2.
Lemma weak_respectful2_respectful2
clo (RES: weak_respectful2 clo):
respectful2 (rclo2 clo).
Proof.
inversion RES. econstructor; [eapply rclo2_mon|]. intros.
induction PR; intros.
- eapply gf_mon; [apply GF, R|]. intros.
apply rclo2_incl. apply PR.
- eapply gf_mon.
+ eapply WEAK_RESPECTFUL0; [|apply H|apply CLOR'].
intros. eapply rclo2_mon; [apply R', PR|apply LE].
+ intros. apply rclo2_rclo2;[apply WEAK_MON0|apply PR].
- eapply gf_mon; [apply CLOR'|].
intros. eapply rclo2_mon; [apply R', PR| apply LE].
Qed.
Lemma upto2_init:
paco2 (compose gf gres2) bot2 <2= paco2 gf bot2.
Proof.
apply sound2_is_gf.
apply respectful2_is_sound2.
apply grespectful2_respectful2.
Qed.
Lemma upto2_final:
paco2 gf <3= paco2 (compose gf gres2).
Proof.
pcofix CIH. intros. _punfold PR; [|apply gf_mon]. pfold.
eapply gf_mon; [|apply grespectful2_incl].
eapply gf_mon; [apply PR|]. intros. right.
inversion PR0; [apply CIH, H | apply CIH0, H].
Qed.
Lemma upto2_step
r clo (RES: weak_respectful2 clo):
clo (paco2 (compose gf gres2) r) <2= paco2 (compose gf gres2) r.
Proof.
intros. apply grespectful2_incl_rev.
assert (RES' := weak_respectful2_respectful2 RES).
eapply grespectful2_greatest; [apply RES'|].
eapply rclo2_base; [apply RES|].
inversion RES. apply PR.
Qed.
Lemma upto2_step_under
r clo (RES: weak_respectful2 clo):
clo (gres2 r) <2= gres2 r.
Proof.
intros. apply weak_respectful2_respectful2 in RES.
eapply grespectful2_compose; [apply RES|].
econstructor 2; [intros; constructor 1; apply PR0 | apply PR].
Qed.
End Respectful2.
Lemma grespectful2_impl T0 T1 (gf gf': rel2 T0 T1 -> rel2 T0 T1) r x0 x1
(PR: gres2 gf r x0 x1)
(EQ: forall r x0 x1, gf r x0 x1 <-> gf' r x0 x1):
gres2 gf' r x0 x1.
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 grespectful2_iff T0 T1 (gf gf': rel2 T0 T1 -> rel2 T0 T1) r x0 x1
(EQ: forall r x0 x1, gf r x0 x1 <-> gf' r x0 x1):
gres2 gf r x0 x1 <-> gres2 gf' r x0 x1.
Proof.
split; intros.
- eapply grespectful2_impl; [apply H | apply EQ].
- eapply grespectful2_impl; [apply H | split; apply EQ].
Qed.
Hint Constructors sound2.
Hint Constructors respectful2.
Hint Constructors gres2.
Hint Resolve gfclo2_mon : paco.
Hint Resolve gfgres2_mon : paco.
Hint Resolve grespectful2_incl.
Hint Resolve rclo2_mon: paco.
Hint Constructors weak_respectful2.
Ltac pupto2_init := eapply upto2_init; [eauto with paco|].
Ltac pupto2_final := first [eapply upto2_final; [eauto with paco|] | eapply grespectful2_incl].
Ltac pupto2 H := first [eapply upto2_step|eapply upto2_step_under]; [|eapply H|]; [eauto with paco|].