Library Paco.paco4_upto
Require Export Program.Basics. Open Scope program_scope.
Require Import paco4.
Set Implicit Arguments.
Section Respectful4.
Variable T0 : Type.
Variable T1 : forall (x0: @T0), Type.
Variable T2 : forall (x0: @T0) (x1: @T1 x0), Type.
Variable T3 : forall (x0: @T0) (x1: @T1 x0) (x2: @T2 x0 x1), Type.
Local Notation rel := (rel4 T0 T1 T2 T3).
Variable gf: rel -> rel.
Hypothesis gf_mon: monotone4 gf.
Inductive sound4 (clo: rel -> rel): Prop :=
| sound4_intro
(MON: monotone4 clo)
(SOUND:
forall r (PFIX: r <4= gf (clo r)),
r <4= paco4 gf bot4)
.
Hint Constructors sound4.
Structure respectful4 (clo: rel -> rel) : Prop :=
respectful4_intro {
MON: monotone4 clo;
RESPECTFUL:
forall l r (LE: l <4= r) (GF: l <4= gf r),
clo l <4= gf (clo r);
}.
Hint Constructors respectful4.
Inductive gres4 (r: rel) e0 e1 e2 e3 : Prop :=
| gres4_intro
clo
(RES: respectful4 clo)
(CLO: clo r e0 e1 e2 e3)
.
Hint Constructors gres4.
Lemma gfclo4_mon: forall clo, sound4 clo -> monotone4 (compose gf clo).
Proof.
intros; destruct H; red; intros.
eapply gf_mon; [apply IN|intros; eapply MON0; [apply PR|apply LE]].
Qed.
Hint Resolve gfclo4_mon : paco.
Lemma sound4_is_gf: forall clo (UPTO: sound4 clo),
paco4 (compose gf clo) bot4 <4= paco4 gf bot4.
Proof.
intros. _punfold PR; [|apply gfclo4_mon, UPTO]. edestruct UPTO.
eapply (SOUND (paco4 (compose gf clo) bot4)).
- intros. _punfold PR0; [|apply gfclo4_mon, UPTO].
eapply (gfclo4_mon UPTO); [apply PR0| intros; destruct PR1; [apply H|destruct H]].
- pfold. apply PR.
Qed.
Lemma respectful4_is_sound4: respectful4 <1= sound4.
Proof.
intro clo.
set (rclo := fix rclo clo n (r: rel) :=
match n with
| 0 => r
| S n' => rclo clo n' r \4/ clo (rclo clo n' r)
end).
intros. destruct PR. econstructor; [apply MON0|].
intros. set (rr e0 e1 e2 e3 := exists n, rclo clo n r e0 e1 e2 e3).
assert (rr x0 x1 x2 x3) by (exists 0; apply PR); clear PR.
cut (forall n, rclo clo n r <4= gf (rclo clo (S n) r)).
{ intro X; revert x0 x1 x2 x3 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 respectful4_compose
clo0 clo1
(RES0: respectful4 clo0)
(RES1: respectful4 clo1):
respectful4 (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 grespectful4_mon: monotone4 gres4.
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 grespectful4_respectful4: respectful4 gres4.
Proof.
econstructor; [apply grespectful4_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 gfgres4_mon: monotone4 (compose gf gres4).
Proof.
destruct grespectful4_respectful4.
unfold monotone4. intros. eapply gf_mon; [eapply IN|].
intros. eapply MON0;[apply PR|apply LE].
Qed.
Hint Resolve gfgres4_mon : paco.
Lemma grespectful4_greatest: forall clo (RES: respectful4 clo), clo <5= gres4.
Proof. intros. econstructor;[apply RES|apply PR]. Qed.
Lemma grespectful4_incl: forall r, r <4= gres4 r.
Proof.
intros; eexists (fun x => x).
- econstructor.
+ red; intros; apply LE, IN.
+ intros; apply GF, PR0.
- apply PR.
Qed.
Hint Resolve grespectful4_incl.
Lemma grespectful4_compose: forall clo (RES: respectful4 clo) r,
clo (gres4 r) <4= gres4 r.
Proof.
intros; eapply grespectful4_greatest with (clo := compose clo gres4); [|apply PR].
apply respectful4_compose; [apply RES|apply grespectful4_respectful4].
Qed.
Lemma grespectful4_incl_rev: forall r,
gres4 (paco4 (compose gf gres4) r) <4= paco4 (compose gf gres4) r.
Proof.
intro r; pcofix CIH; intros; pfold.
eapply gf_mon, grespectful4_compose, grespectful4_respectful4.
destruct grespectful4_respectful4; eapply RESPECTFUL0, PR; intros; [apply grespectful4_incl; right; apply CIH, grespectful4_incl, PR0|].
_punfold PR0; [|apply gfgres4_mon].
eapply gfgres4_mon; [apply PR0|].
intros; destruct PR1.
- left. eapply paco4_mon; [apply H| apply CIH0].
- right. eapply CIH0, H.
Qed.
Inductive rclo4 clo (r: rel): rel :=
| rclo4_incl
e0 e1 e2 e3
(R: r e0 e1 e2 e3):
@rclo4 clo r e0 e1 e2 e3
| rclo4_step'
r' e0 e1 e2 e3
(R': r' <4= rclo4 clo r)
(CLOR':clo r' e0 e1 e2 e3):
@rclo4 clo r e0 e1 e2 e3
| rclo4_gf
r' e0 e1 e2 e3
(R': r' <4= rclo4 clo r)
(CLOR':@gf r' e0 e1 e2 e3):
@rclo4 clo r e0 e1 e2 e3
.
Lemma rclo4_mon clo:
monotone4 (rclo4 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 rclo4_mon: paco.
Lemma rclo4_base
clo
(MON: monotone4 clo):
clo <5= rclo4 clo.
Proof.
intros. econstructor 2; [intros; apply PR0|].
eapply MON; [apply PR|intros; constructor; apply PR0].
Qed.
Lemma rclo4_step
(clo: rel -> rel) r:
clo (rclo4 clo r) <4= rclo4 clo r.
Proof.
intros. econstructor 2; [intros; apply PR0|apply PR].
Qed.
Lemma rclo4_rclo4
clo r
(MON: monotone4 clo):
rclo4 clo (rclo4 clo r) <4= rclo4 clo r.
Proof.
intros. induction PR.
- eapply R.
- econstructor 2; [eapply H | eapply CLOR'].
- econstructor 3; [eapply H | eapply CLOR'].
Qed.
Structure weak_respectful4 (clo: rel -> rel) : Prop :=
weak_respectful4_intro {
WEAK_MON: monotone4 clo;
WEAK_RESPECTFUL:
forall l r (LE: l <4= r) (GF: l <4= gf r),
clo l <4= gf (rclo4 clo r);
}.
Hint Constructors weak_respectful4.
Lemma weak_respectful4_respectful4
clo (RES: weak_respectful4 clo):
respectful4 (rclo4 clo).
Proof.
inversion RES. econstructor; [eapply rclo4_mon|]. intros.
induction PR; intros.
- eapply gf_mon; [apply GF, R|]. intros.
apply rclo4_incl. apply PR.
- eapply gf_mon.
+ eapply WEAK_RESPECTFUL0; [|apply H|apply CLOR'].
intros. eapply rclo4_mon; [apply R', PR|apply LE].
+ intros. apply rclo4_rclo4;[apply WEAK_MON0|apply PR].
- eapply gf_mon; [apply CLOR'|].
intros. eapply rclo4_mon; [apply R', PR| apply LE].
Qed.
Lemma upto4_init:
paco4 (compose gf gres4) bot4 <4= paco4 gf bot4.
Proof.
apply sound4_is_gf.
apply respectful4_is_sound4.
apply grespectful4_respectful4.
Qed.
Lemma upto4_final:
paco4 gf <5= paco4 (compose gf gres4).
Proof.
pcofix CIH. intros. _punfold PR; [|apply gf_mon]. pfold.
eapply gf_mon; [|apply grespectful4_incl].
eapply gf_mon; [apply PR|]. intros. right.
inversion PR0; [apply CIH, H | apply CIH0, H].
Qed.
Lemma upto4_step
r clo (RES: weak_respectful4 clo):
clo (paco4 (compose gf gres4) r) <4= paco4 (compose gf gres4) r.
Proof.
intros. apply grespectful4_incl_rev.
assert (RES' := weak_respectful4_respectful4 RES).
eapply grespectful4_greatest; [apply RES'|].
eapply rclo4_base; [apply RES|].
inversion RES. apply PR.
Qed.
Lemma upto4_step_under
r clo (RES: weak_respectful4 clo):
clo (gres4 r) <4= gres4 r.
Proof.
intros. apply weak_respectful4_respectful4 in RES.
eapply grespectful4_compose; [apply RES|].
econstructor 2; [intros; constructor 1; apply PR0 | apply PR].
Qed.
End Respectful4.
Lemma grespectful4_impl T0 T1 T2 T3 (gf gf': rel4 T0 T1 T2 T3 -> rel4 T0 T1 T2 T3) r x0 x1 x2 x3
(PR: gres4 gf r x0 x1 x2 x3)
(EQ: forall r x0 x1 x2 x3, gf r x0 x1 x2 x3 <-> gf' r x0 x1 x2 x3):
gres4 gf' r x0 x1 x2 x3.
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 grespectful4_iff T0 T1 T2 T3 (gf gf': rel4 T0 T1 T2 T3 -> rel4 T0 T1 T2 T3) r x0 x1 x2 x3
(EQ: forall r x0 x1 x2 x3, gf r x0 x1 x2 x3 <-> gf' r x0 x1 x2 x3):
gres4 gf r x0 x1 x2 x3 <-> gres4 gf' r x0 x1 x2 x3.
Proof.
split; intros.
- eapply grespectful4_impl; [apply H | apply EQ].
- eapply grespectful4_impl; [apply H | split; apply EQ].
Qed.
Hint Constructors sound4.
Hint Constructors respectful4.
Hint Constructors gres4.
Hint Resolve gfclo4_mon : paco.
Hint Resolve gfgres4_mon : paco.
Hint Resolve grespectful4_incl.
Hint Resolve rclo4_mon: paco.
Hint Constructors weak_respectful4.
Ltac pupto4_init := eapply upto4_init; [eauto with paco|].
Ltac pupto4_final := first [eapply upto4_final; [eauto with paco|] | eapply grespectful4_incl].
Ltac pupto4 H := first [eapply upto4_step|eapply upto4_step_under]; [|eapply H|]; [eauto with paco|].
Require Import paco4.
Set Implicit Arguments.
Section Respectful4.
Variable T0 : Type.
Variable T1 : forall (x0: @T0), Type.
Variable T2 : forall (x0: @T0) (x1: @T1 x0), Type.
Variable T3 : forall (x0: @T0) (x1: @T1 x0) (x2: @T2 x0 x1), Type.
Local Notation rel := (rel4 T0 T1 T2 T3).
Variable gf: rel -> rel.
Hypothesis gf_mon: monotone4 gf.
Inductive sound4 (clo: rel -> rel): Prop :=
| sound4_intro
(MON: monotone4 clo)
(SOUND:
forall r (PFIX: r <4= gf (clo r)),
r <4= paco4 gf bot4)
.
Hint Constructors sound4.
Structure respectful4 (clo: rel -> rel) : Prop :=
respectful4_intro {
MON: monotone4 clo;
RESPECTFUL:
forall l r (LE: l <4= r) (GF: l <4= gf r),
clo l <4= gf (clo r);
}.
Hint Constructors respectful4.
Inductive gres4 (r: rel) e0 e1 e2 e3 : Prop :=
| gres4_intro
clo
(RES: respectful4 clo)
(CLO: clo r e0 e1 e2 e3)
.
Hint Constructors gres4.
Lemma gfclo4_mon: forall clo, sound4 clo -> monotone4 (compose gf clo).
Proof.
intros; destruct H; red; intros.
eapply gf_mon; [apply IN|intros; eapply MON0; [apply PR|apply LE]].
Qed.
Hint Resolve gfclo4_mon : paco.
Lemma sound4_is_gf: forall clo (UPTO: sound4 clo),
paco4 (compose gf clo) bot4 <4= paco4 gf bot4.
Proof.
intros. _punfold PR; [|apply gfclo4_mon, UPTO]. edestruct UPTO.
eapply (SOUND (paco4 (compose gf clo) bot4)).
- intros. _punfold PR0; [|apply gfclo4_mon, UPTO].
eapply (gfclo4_mon UPTO); [apply PR0| intros; destruct PR1; [apply H|destruct H]].
- pfold. apply PR.
Qed.
Lemma respectful4_is_sound4: respectful4 <1= sound4.
Proof.
intro clo.
set (rclo := fix rclo clo n (r: rel) :=
match n with
| 0 => r
| S n' => rclo clo n' r \4/ clo (rclo clo n' r)
end).
intros. destruct PR. econstructor; [apply MON0|].
intros. set (rr e0 e1 e2 e3 := exists n, rclo clo n r e0 e1 e2 e3).
assert (rr x0 x1 x2 x3) by (exists 0; apply PR); clear PR.
cut (forall n, rclo clo n r <4= gf (rclo clo (S n) r)).
{ intro X; revert x0 x1 x2 x3 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 respectful4_compose
clo0 clo1
(RES0: respectful4 clo0)
(RES1: respectful4 clo1):
respectful4 (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 grespectful4_mon: monotone4 gres4.
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 grespectful4_respectful4: respectful4 gres4.
Proof.
econstructor; [apply grespectful4_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 gfgres4_mon: monotone4 (compose gf gres4).
Proof.
destruct grespectful4_respectful4.
unfold monotone4. intros. eapply gf_mon; [eapply IN|].
intros. eapply MON0;[apply PR|apply LE].
Qed.
Hint Resolve gfgres4_mon : paco.
Lemma grespectful4_greatest: forall clo (RES: respectful4 clo), clo <5= gres4.
Proof. intros. econstructor;[apply RES|apply PR]. Qed.
Lemma grespectful4_incl: forall r, r <4= gres4 r.
Proof.
intros; eexists (fun x => x).
- econstructor.
+ red; intros; apply LE, IN.
+ intros; apply GF, PR0.
- apply PR.
Qed.
Hint Resolve grespectful4_incl.
Lemma grespectful4_compose: forall clo (RES: respectful4 clo) r,
clo (gres4 r) <4= gres4 r.
Proof.
intros; eapply grespectful4_greatest with (clo := compose clo gres4); [|apply PR].
apply respectful4_compose; [apply RES|apply grespectful4_respectful4].
Qed.
Lemma grespectful4_incl_rev: forall r,
gres4 (paco4 (compose gf gres4) r) <4= paco4 (compose gf gres4) r.
Proof.
intro r; pcofix CIH; intros; pfold.
eapply gf_mon, grespectful4_compose, grespectful4_respectful4.
destruct grespectful4_respectful4; eapply RESPECTFUL0, PR; intros; [apply grespectful4_incl; right; apply CIH, grespectful4_incl, PR0|].
_punfold PR0; [|apply gfgres4_mon].
eapply gfgres4_mon; [apply PR0|].
intros; destruct PR1.
- left. eapply paco4_mon; [apply H| apply CIH0].
- right. eapply CIH0, H.
Qed.
Inductive rclo4 clo (r: rel): rel :=
| rclo4_incl
e0 e1 e2 e3
(R: r e0 e1 e2 e3):
@rclo4 clo r e0 e1 e2 e3
| rclo4_step'
r' e0 e1 e2 e3
(R': r' <4= rclo4 clo r)
(CLOR':clo r' e0 e1 e2 e3):
@rclo4 clo r e0 e1 e2 e3
| rclo4_gf
r' e0 e1 e2 e3
(R': r' <4= rclo4 clo r)
(CLOR':@gf r' e0 e1 e2 e3):
@rclo4 clo r e0 e1 e2 e3
.
Lemma rclo4_mon clo:
monotone4 (rclo4 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 rclo4_mon: paco.
Lemma rclo4_base
clo
(MON: monotone4 clo):
clo <5= rclo4 clo.
Proof.
intros. econstructor 2; [intros; apply PR0|].
eapply MON; [apply PR|intros; constructor; apply PR0].
Qed.
Lemma rclo4_step
(clo: rel -> rel) r:
clo (rclo4 clo r) <4= rclo4 clo r.
Proof.
intros. econstructor 2; [intros; apply PR0|apply PR].
Qed.
Lemma rclo4_rclo4
clo r
(MON: monotone4 clo):
rclo4 clo (rclo4 clo r) <4= rclo4 clo r.
Proof.
intros. induction PR.
- eapply R.
- econstructor 2; [eapply H | eapply CLOR'].
- econstructor 3; [eapply H | eapply CLOR'].
Qed.
Structure weak_respectful4 (clo: rel -> rel) : Prop :=
weak_respectful4_intro {
WEAK_MON: monotone4 clo;
WEAK_RESPECTFUL:
forall l r (LE: l <4= r) (GF: l <4= gf r),
clo l <4= gf (rclo4 clo r);
}.
Hint Constructors weak_respectful4.
Lemma weak_respectful4_respectful4
clo (RES: weak_respectful4 clo):
respectful4 (rclo4 clo).
Proof.
inversion RES. econstructor; [eapply rclo4_mon|]. intros.
induction PR; intros.
- eapply gf_mon; [apply GF, R|]. intros.
apply rclo4_incl. apply PR.
- eapply gf_mon.
+ eapply WEAK_RESPECTFUL0; [|apply H|apply CLOR'].
intros. eapply rclo4_mon; [apply R', PR|apply LE].
+ intros. apply rclo4_rclo4;[apply WEAK_MON0|apply PR].
- eapply gf_mon; [apply CLOR'|].
intros. eapply rclo4_mon; [apply R', PR| apply LE].
Qed.
Lemma upto4_init:
paco4 (compose gf gres4) bot4 <4= paco4 gf bot4.
Proof.
apply sound4_is_gf.
apply respectful4_is_sound4.
apply grespectful4_respectful4.
Qed.
Lemma upto4_final:
paco4 gf <5= paco4 (compose gf gres4).
Proof.
pcofix CIH. intros. _punfold PR; [|apply gf_mon]. pfold.
eapply gf_mon; [|apply grespectful4_incl].
eapply gf_mon; [apply PR|]. intros. right.
inversion PR0; [apply CIH, H | apply CIH0, H].
Qed.
Lemma upto4_step
r clo (RES: weak_respectful4 clo):
clo (paco4 (compose gf gres4) r) <4= paco4 (compose gf gres4) r.
Proof.
intros. apply grespectful4_incl_rev.
assert (RES' := weak_respectful4_respectful4 RES).
eapply grespectful4_greatest; [apply RES'|].
eapply rclo4_base; [apply RES|].
inversion RES. apply PR.
Qed.
Lemma upto4_step_under
r clo (RES: weak_respectful4 clo):
clo (gres4 r) <4= gres4 r.
Proof.
intros. apply weak_respectful4_respectful4 in RES.
eapply grespectful4_compose; [apply RES|].
econstructor 2; [intros; constructor 1; apply PR0 | apply PR].
Qed.
End Respectful4.
Lemma grespectful4_impl T0 T1 T2 T3 (gf gf': rel4 T0 T1 T2 T3 -> rel4 T0 T1 T2 T3) r x0 x1 x2 x3
(PR: gres4 gf r x0 x1 x2 x3)
(EQ: forall r x0 x1 x2 x3, gf r x0 x1 x2 x3 <-> gf' r x0 x1 x2 x3):
gres4 gf' r x0 x1 x2 x3.
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 grespectful4_iff T0 T1 T2 T3 (gf gf': rel4 T0 T1 T2 T3 -> rel4 T0 T1 T2 T3) r x0 x1 x2 x3
(EQ: forall r x0 x1 x2 x3, gf r x0 x1 x2 x3 <-> gf' r x0 x1 x2 x3):
gres4 gf r x0 x1 x2 x3 <-> gres4 gf' r x0 x1 x2 x3.
Proof.
split; intros.
- eapply grespectful4_impl; [apply H | apply EQ].
- eapply grespectful4_impl; [apply H | split; apply EQ].
Qed.
Hint Constructors sound4.
Hint Constructors respectful4.
Hint Constructors gres4.
Hint Resolve gfclo4_mon : paco.
Hint Resolve gfgres4_mon : paco.
Hint Resolve grespectful4_incl.
Hint Resolve rclo4_mon: paco.
Hint Constructors weak_respectful4.
Ltac pupto4_init := eapply upto4_init; [eauto with paco|].
Ltac pupto4_final := first [eapply upto4_final; [eauto with paco|] | eapply grespectful4_incl].
Ltac pupto4 H := first [eapply upto4_step|eapply upto4_step_under]; [|eapply H|]; [eauto with paco|].