Library Paco.paco1_upto
Require Export Program.Basics. Open Scope program_scope.
Require Import paco1.
Set Implicit Arguments.
Section Respectful1.
Variable T0 : Type.
Local Notation rel := (rel1 T0).
Variable gf: rel -> rel.
Hypothesis gf_mon: monotone1 gf.
Inductive sound1 (clo: rel -> rel): Prop :=
| sound1_intro
(MON: monotone1 clo)
(SOUND:
forall r (PFIX: r <1= gf (clo r)),
r <1= paco1 gf bot1)
.
Hint Constructors sound1.
Structure respectful1 (clo: rel -> rel) : Prop :=
respectful1_intro {
MON: monotone1 clo;
RESPECTFUL:
forall l r (LE: l <1= r) (GF: l <1= gf r),
clo l <1= gf (clo r);
}.
Hint Constructors respectful1.
Inductive gres1 (r: rel) e0 : Prop :=
| gres1_intro
clo
(RES: respectful1 clo)
(CLO: clo r e0)
.
Hint Constructors gres1.
Lemma gfclo1_mon: forall clo, sound1 clo -> monotone1 (compose gf clo).
Proof.
intros; destruct H; red; intros.
eapply gf_mon; [apply IN|intros; eapply MON0; [apply PR|apply LE]].
Qed.
Hint Resolve gfclo1_mon : paco.
Lemma sound1_is_gf: forall clo (UPTO: sound1 clo),
paco1 (compose gf clo) bot1 <1= paco1 gf bot1.
Proof.
intros. _punfold PR; [|apply gfclo1_mon, UPTO]. edestruct UPTO.
eapply (SOUND (paco1 (compose gf clo) bot1)).
- intros. _punfold PR0; [|apply gfclo1_mon, UPTO].
eapply (gfclo1_mon UPTO); [apply PR0| intros; destruct PR1; [apply H|destruct H]].
- pfold. apply PR.
Qed.
Lemma respectful1_is_sound1: respectful1 <1= sound1.
Proof.
intro clo.
set (rclo := fix rclo clo n (r: rel) :=
match n with
| 0 => r
| S n' => rclo clo n' r \1/ clo (rclo clo n' r)
end).
intros. destruct PR. econstructor; [apply MON0|].
intros. set (rr e0 := exists n, rclo clo n r e0).
assert (rr x0) by (exists 0; apply PR); clear PR.
cut (forall n, rclo clo n r <1= gf (rclo clo (S n) r)).
{ intro X; revert x0 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 respectful1_compose
clo0 clo1
(RES0: respectful1 clo0)
(RES1: respectful1 clo1):
respectful1 (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 grespectful1_mon: monotone1 gres1.
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 grespectful1_respectful1: respectful1 gres1.
Proof.
econstructor; [apply grespectful1_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 gfgres1_mon: monotone1 (compose gf gres1).
Proof.
destruct grespectful1_respectful1.
unfold monotone1. intros. eapply gf_mon; [eapply IN|].
intros. eapply MON0;[apply PR|apply LE].
Qed.
Hint Resolve gfgres1_mon : paco.
Lemma grespectful1_greatest: forall clo (RES: respectful1 clo), clo <2= gres1.
Proof. intros. econstructor;[apply RES|apply PR]. Qed.
Lemma grespectful1_incl: forall r, r <1= gres1 r.
Proof.
intros; eexists (fun x => x).
- econstructor.
+ red; intros; apply LE, IN.
+ intros; apply GF, PR0.
- apply PR.
Qed.
Hint Resolve grespectful1_incl.
Lemma grespectful1_compose: forall clo (RES: respectful1 clo) r,
clo (gres1 r) <1= gres1 r.
Proof.
intros; eapply grespectful1_greatest with (clo := compose clo gres1); [|apply PR].
apply respectful1_compose; [apply RES|apply grespectful1_respectful1].
Qed.
Lemma grespectful1_incl_rev: forall r,
gres1 (paco1 (compose gf gres1) r) <1= paco1 (compose gf gres1) r.
Proof.
intro r; pcofix CIH; intros; pfold.
eapply gf_mon, grespectful1_compose, grespectful1_respectful1.
destruct grespectful1_respectful1; eapply RESPECTFUL0, PR; intros; [apply grespectful1_incl; right; apply CIH, grespectful1_incl, PR0|].
_punfold PR0; [|apply gfgres1_mon].
eapply gfgres1_mon; [apply PR0|].
intros; destruct PR1.
- left. eapply paco1_mon; [apply H| apply CIH0].
- right. eapply CIH0, H.
Qed.
Inductive rclo1 clo (r: rel): rel :=
| rclo1_incl
e0
(R: r e0):
@rclo1 clo r e0
| rclo1_step'
r' e0
(R': r' <1= rclo1 clo r)
(CLOR':clo r' e0):
@rclo1 clo r e0
| rclo1_gf
r' e0
(R': r' <1= rclo1 clo r)
(CLOR':@gf r' e0):
@rclo1 clo r e0
.
Lemma rclo1_mon clo:
monotone1 (rclo1 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 rclo1_mon: paco.
Lemma rclo1_base
clo
(MON: monotone1 clo):
clo <2= rclo1 clo.
Proof.
intros. econstructor 2; [intros; apply PR0|].
eapply MON; [apply PR|intros; constructor; apply PR0].
Qed.
Lemma rclo1_step
(clo: rel -> rel) r:
clo (rclo1 clo r) <1= rclo1 clo r.
Proof.
intros. econstructor 2; [intros; apply PR0|apply PR].
Qed.
Lemma rclo1_rclo1
clo r
(MON: monotone1 clo):
rclo1 clo (rclo1 clo r) <1= rclo1 clo r.
Proof.
intros. induction PR.
- eapply R.
- econstructor 2; [eapply H | eapply CLOR'].
- econstructor 3; [eapply H | eapply CLOR'].
Qed.
Structure weak_respectful1 (clo: rel -> rel) : Prop :=
weak_respectful1_intro {
WEAK_MON: monotone1 clo;
WEAK_RESPECTFUL:
forall l r (LE: l <1= r) (GF: l <1= gf r),
clo l <1= gf (rclo1 clo r);
}.
Hint Constructors weak_respectful1.
Lemma weak_respectful1_respectful1
clo (RES: weak_respectful1 clo):
respectful1 (rclo1 clo).
Proof.
inversion RES. econstructor; [eapply rclo1_mon|]. intros.
induction PR; intros.
- eapply gf_mon; [apply GF, R|]. intros.
apply rclo1_incl. apply PR.
- eapply gf_mon.
+ eapply WEAK_RESPECTFUL0; [|apply H|apply CLOR'].
intros. eapply rclo1_mon; [apply R', PR|apply LE].
+ intros. apply rclo1_rclo1;[apply WEAK_MON0|apply PR].
- eapply gf_mon; [apply CLOR'|].
intros. eapply rclo1_mon; [apply R', PR| apply LE].
Qed.
Lemma upto1_init:
paco1 (compose gf gres1) bot1 <1= paco1 gf bot1.
Proof.
apply sound1_is_gf.
apply respectful1_is_sound1.
apply grespectful1_respectful1.
Qed.
Lemma upto1_final:
paco1 gf <2= paco1 (compose gf gres1).
Proof.
pcofix CIH. intros. _punfold PR; [|apply gf_mon]. pfold.
eapply gf_mon; [|apply grespectful1_incl].
eapply gf_mon; [apply PR|]. intros. right.
inversion PR0; [apply CIH, H | apply CIH0, H].
Qed.
Lemma upto1_step
r clo (RES: weak_respectful1 clo):
clo (paco1 (compose gf gres1) r) <1= paco1 (compose gf gres1) r.
Proof.
intros. apply grespectful1_incl_rev.
assert (RES' := weak_respectful1_respectful1 RES).
eapply grespectful1_greatest; [apply RES'|].
eapply rclo1_base; [apply RES|].
inversion RES. apply PR.
Qed.
Lemma upto1_step_under
r clo (RES: weak_respectful1 clo):
clo (gres1 r) <1= gres1 r.
Proof.
intros. apply weak_respectful1_respectful1 in RES.
eapply grespectful1_compose; [apply RES|].
econstructor 2; [intros; constructor 1; apply PR0 | apply PR].
Qed.
End Respectful1.
Lemma grespectful1_impl T0 (gf gf': rel1 T0 -> rel1 T0) r x0
(PR: gres1 gf r x0)
(EQ: forall r x0, gf r x0 <-> gf' r x0):
gres1 gf' r x0.
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 grespectful1_iff T0 (gf gf': rel1 T0 -> rel1 T0) r x0
(EQ: forall r x0, gf r x0 <-> gf' r x0):
gres1 gf r x0 <-> gres1 gf' r x0.
Proof.
split; intros.
- eapply grespectful1_impl; [apply H | apply EQ].
- eapply grespectful1_impl; [apply H | split; apply EQ].
Qed.
Hint Constructors sound1.
Hint Constructors respectful1.
Hint Constructors gres1.
Hint Resolve gfclo1_mon : paco.
Hint Resolve gfgres1_mon : paco.
Hint Resolve grespectful1_incl.
Hint Resolve rclo1_mon: paco.
Hint Constructors weak_respectful1.
Ltac pupto1_init := eapply upto1_init; [eauto with paco|].
Ltac pupto1_final := first [eapply upto1_final; [eauto with paco|] | eapply grespectful1_incl].
Ltac pupto1 H := first [eapply upto1_step|eapply upto1_step_under]; [|eapply H|]; [eauto with paco|].
Require Import paco1.
Set Implicit Arguments.
Section Respectful1.
Variable T0 : Type.
Local Notation rel := (rel1 T0).
Variable gf: rel -> rel.
Hypothesis gf_mon: monotone1 gf.
Inductive sound1 (clo: rel -> rel): Prop :=
| sound1_intro
(MON: monotone1 clo)
(SOUND:
forall r (PFIX: r <1= gf (clo r)),
r <1= paco1 gf bot1)
.
Hint Constructors sound1.
Structure respectful1 (clo: rel -> rel) : Prop :=
respectful1_intro {
MON: monotone1 clo;
RESPECTFUL:
forall l r (LE: l <1= r) (GF: l <1= gf r),
clo l <1= gf (clo r);
}.
Hint Constructors respectful1.
Inductive gres1 (r: rel) e0 : Prop :=
| gres1_intro
clo
(RES: respectful1 clo)
(CLO: clo r e0)
.
Hint Constructors gres1.
Lemma gfclo1_mon: forall clo, sound1 clo -> monotone1 (compose gf clo).
Proof.
intros; destruct H; red; intros.
eapply gf_mon; [apply IN|intros; eapply MON0; [apply PR|apply LE]].
Qed.
Hint Resolve gfclo1_mon : paco.
Lemma sound1_is_gf: forall clo (UPTO: sound1 clo),
paco1 (compose gf clo) bot1 <1= paco1 gf bot1.
Proof.
intros. _punfold PR; [|apply gfclo1_mon, UPTO]. edestruct UPTO.
eapply (SOUND (paco1 (compose gf clo) bot1)).
- intros. _punfold PR0; [|apply gfclo1_mon, UPTO].
eapply (gfclo1_mon UPTO); [apply PR0| intros; destruct PR1; [apply H|destruct H]].
- pfold. apply PR.
Qed.
Lemma respectful1_is_sound1: respectful1 <1= sound1.
Proof.
intro clo.
set (rclo := fix rclo clo n (r: rel) :=
match n with
| 0 => r
| S n' => rclo clo n' r \1/ clo (rclo clo n' r)
end).
intros. destruct PR. econstructor; [apply MON0|].
intros. set (rr e0 := exists n, rclo clo n r e0).
assert (rr x0) by (exists 0; apply PR); clear PR.
cut (forall n, rclo clo n r <1= gf (rclo clo (S n) r)).
{ intro X; revert x0 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 respectful1_compose
clo0 clo1
(RES0: respectful1 clo0)
(RES1: respectful1 clo1):
respectful1 (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 grespectful1_mon: monotone1 gres1.
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 grespectful1_respectful1: respectful1 gres1.
Proof.
econstructor; [apply grespectful1_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 gfgres1_mon: monotone1 (compose gf gres1).
Proof.
destruct grespectful1_respectful1.
unfold monotone1. intros. eapply gf_mon; [eapply IN|].
intros. eapply MON0;[apply PR|apply LE].
Qed.
Hint Resolve gfgres1_mon : paco.
Lemma grespectful1_greatest: forall clo (RES: respectful1 clo), clo <2= gres1.
Proof. intros. econstructor;[apply RES|apply PR]. Qed.
Lemma grespectful1_incl: forall r, r <1= gres1 r.
Proof.
intros; eexists (fun x => x).
- econstructor.
+ red; intros; apply LE, IN.
+ intros; apply GF, PR0.
- apply PR.
Qed.
Hint Resolve grespectful1_incl.
Lemma grespectful1_compose: forall clo (RES: respectful1 clo) r,
clo (gres1 r) <1= gres1 r.
Proof.
intros; eapply grespectful1_greatest with (clo := compose clo gres1); [|apply PR].
apply respectful1_compose; [apply RES|apply grespectful1_respectful1].
Qed.
Lemma grespectful1_incl_rev: forall r,
gres1 (paco1 (compose gf gres1) r) <1= paco1 (compose gf gres1) r.
Proof.
intro r; pcofix CIH; intros; pfold.
eapply gf_mon, grespectful1_compose, grespectful1_respectful1.
destruct grespectful1_respectful1; eapply RESPECTFUL0, PR; intros; [apply grespectful1_incl; right; apply CIH, grespectful1_incl, PR0|].
_punfold PR0; [|apply gfgres1_mon].
eapply gfgres1_mon; [apply PR0|].
intros; destruct PR1.
- left. eapply paco1_mon; [apply H| apply CIH0].
- right. eapply CIH0, H.
Qed.
Inductive rclo1 clo (r: rel): rel :=
| rclo1_incl
e0
(R: r e0):
@rclo1 clo r e0
| rclo1_step'
r' e0
(R': r' <1= rclo1 clo r)
(CLOR':clo r' e0):
@rclo1 clo r e0
| rclo1_gf
r' e0
(R': r' <1= rclo1 clo r)
(CLOR':@gf r' e0):
@rclo1 clo r e0
.
Lemma rclo1_mon clo:
monotone1 (rclo1 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 rclo1_mon: paco.
Lemma rclo1_base
clo
(MON: monotone1 clo):
clo <2= rclo1 clo.
Proof.
intros. econstructor 2; [intros; apply PR0|].
eapply MON; [apply PR|intros; constructor; apply PR0].
Qed.
Lemma rclo1_step
(clo: rel -> rel) r:
clo (rclo1 clo r) <1= rclo1 clo r.
Proof.
intros. econstructor 2; [intros; apply PR0|apply PR].
Qed.
Lemma rclo1_rclo1
clo r
(MON: monotone1 clo):
rclo1 clo (rclo1 clo r) <1= rclo1 clo r.
Proof.
intros. induction PR.
- eapply R.
- econstructor 2; [eapply H | eapply CLOR'].
- econstructor 3; [eapply H | eapply CLOR'].
Qed.
Structure weak_respectful1 (clo: rel -> rel) : Prop :=
weak_respectful1_intro {
WEAK_MON: monotone1 clo;
WEAK_RESPECTFUL:
forall l r (LE: l <1= r) (GF: l <1= gf r),
clo l <1= gf (rclo1 clo r);
}.
Hint Constructors weak_respectful1.
Lemma weak_respectful1_respectful1
clo (RES: weak_respectful1 clo):
respectful1 (rclo1 clo).
Proof.
inversion RES. econstructor; [eapply rclo1_mon|]. intros.
induction PR; intros.
- eapply gf_mon; [apply GF, R|]. intros.
apply rclo1_incl. apply PR.
- eapply gf_mon.
+ eapply WEAK_RESPECTFUL0; [|apply H|apply CLOR'].
intros. eapply rclo1_mon; [apply R', PR|apply LE].
+ intros. apply rclo1_rclo1;[apply WEAK_MON0|apply PR].
- eapply gf_mon; [apply CLOR'|].
intros. eapply rclo1_mon; [apply R', PR| apply LE].
Qed.
Lemma upto1_init:
paco1 (compose gf gres1) bot1 <1= paco1 gf bot1.
Proof.
apply sound1_is_gf.
apply respectful1_is_sound1.
apply grespectful1_respectful1.
Qed.
Lemma upto1_final:
paco1 gf <2= paco1 (compose gf gres1).
Proof.
pcofix CIH. intros. _punfold PR; [|apply gf_mon]. pfold.
eapply gf_mon; [|apply grespectful1_incl].
eapply gf_mon; [apply PR|]. intros. right.
inversion PR0; [apply CIH, H | apply CIH0, H].
Qed.
Lemma upto1_step
r clo (RES: weak_respectful1 clo):
clo (paco1 (compose gf gres1) r) <1= paco1 (compose gf gres1) r.
Proof.
intros. apply grespectful1_incl_rev.
assert (RES' := weak_respectful1_respectful1 RES).
eapply grespectful1_greatest; [apply RES'|].
eapply rclo1_base; [apply RES|].
inversion RES. apply PR.
Qed.
Lemma upto1_step_under
r clo (RES: weak_respectful1 clo):
clo (gres1 r) <1= gres1 r.
Proof.
intros. apply weak_respectful1_respectful1 in RES.
eapply grespectful1_compose; [apply RES|].
econstructor 2; [intros; constructor 1; apply PR0 | apply PR].
Qed.
End Respectful1.
Lemma grespectful1_impl T0 (gf gf': rel1 T0 -> rel1 T0) r x0
(PR: gres1 gf r x0)
(EQ: forall r x0, gf r x0 <-> gf' r x0):
gres1 gf' r x0.
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 grespectful1_iff T0 (gf gf': rel1 T0 -> rel1 T0) r x0
(EQ: forall r x0, gf r x0 <-> gf' r x0):
gres1 gf r x0 <-> gres1 gf' r x0.
Proof.
split; intros.
- eapply grespectful1_impl; [apply H | apply EQ].
- eapply grespectful1_impl; [apply H | split; apply EQ].
Qed.
Hint Constructors sound1.
Hint Constructors respectful1.
Hint Constructors gres1.
Hint Resolve gfclo1_mon : paco.
Hint Resolve gfgres1_mon : paco.
Hint Resolve grespectful1_incl.
Hint Resolve rclo1_mon: paco.
Hint Constructors weak_respectful1.
Ltac pupto1_init := eapply upto1_init; [eauto with paco|].
Ltac pupto1_final := first [eapply upto1_final; [eauto with paco|] | eapply grespectful1_incl].
Ltac pupto1 H := first [eapply upto1_step|eapply upto1_step_under]; [|eapply H|]; [eauto with paco|].