Library Paco.paco5_upto
Require Export Program.Basics. Open Scope program_scope.
Require Import paco5.
Set Implicit Arguments.
Section Respectful5.
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.
Variable T4 : forall (x0: @T0) (x1: @T1 x0) (x2: @T2 x0 x1) (x3: @T3 x0 x1 x2), Type.
Local Notation rel := (rel5 T0 T1 T2 T3 T4).
Variable gf: rel -> rel.
Hypothesis gf_mon: monotone5 gf.
Inductive sound5 (clo: rel -> rel): Prop :=
| sound5_intro
(MON: monotone5 clo)
(SOUND:
forall r (PFIX: r <5= gf (clo r)),
r <5= paco5 gf bot5)
.
Hint Constructors sound5.
Structure respectful5 (clo: rel -> rel) : Prop :=
respectful5_intro {
MON: monotone5 clo;
RESPECTFUL:
forall l r (LE: l <5= r) (GF: l <5= gf r),
clo l <5= gf (clo r);
}.
Hint Constructors respectful5.
Inductive gres5 (r: rel) e0 e1 e2 e3 e4 : Prop :=
| gres5_intro
clo
(RES: respectful5 clo)
(CLO: clo r e0 e1 e2 e3 e4)
.
Hint Constructors gres5.
Lemma gfclo5_mon: forall clo, sound5 clo -> monotone5 (compose gf clo).
Proof.
intros; destruct H; red; intros.
eapply gf_mon; [apply IN|intros; eapply MON0; [apply PR|apply LE]].
Qed.
Hint Resolve gfclo5_mon : paco.
Lemma sound5_is_gf: forall clo (UPTO: sound5 clo),
paco5 (compose gf clo) bot5 <5= paco5 gf bot5.
Proof.
intros. _punfold PR; [|apply gfclo5_mon, UPTO]. edestruct UPTO.
eapply (SOUND (paco5 (compose gf clo) bot5)).
- intros. _punfold PR0; [|apply gfclo5_mon, UPTO].
eapply (gfclo5_mon UPTO); [apply PR0| intros; destruct PR1; [apply H|destruct H]].
- pfold. apply PR.
Qed.
Lemma respectful5_is_sound5: respectful5 <1= sound5.
Proof.
intro clo.
set (rclo := fix rclo clo n (r: rel) :=
match n with
| 0 => r
| S n' => rclo clo n' r \5/ clo (rclo clo n' r)
end).
intros. destruct PR. econstructor; [apply MON0|].
intros. set (rr e0 e1 e2 e3 e4 := exists n, rclo clo n r e0 e1 e2 e3 e4).
assert (rr x0 x1 x2 x3 x4) by (exists 0; apply PR); clear PR.
cut (forall n, rclo clo n r <5= gf (rclo clo (S n) r)).
{ intro X; revert x0 x1 x2 x3 x4 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 respectful5_compose
clo0 clo1
(RES0: respectful5 clo0)
(RES1: respectful5 clo1):
respectful5 (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 grespectful5_mon: monotone5 gres5.
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 grespectful5_respectful5: respectful5 gres5.
Proof.
econstructor; [apply grespectful5_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 gfgres5_mon: monotone5 (compose gf gres5).
Proof.
destruct grespectful5_respectful5.
unfold monotone5. intros. eapply gf_mon; [eapply IN|].
intros. eapply MON0;[apply PR|apply LE].
Qed.
Hint Resolve gfgres5_mon : paco.
Lemma grespectful5_greatest: forall clo (RES: respectful5 clo), clo <6= gres5.
Proof. intros. econstructor;[apply RES|apply PR]. Qed.
Lemma grespectful5_incl: forall r, r <5= gres5 r.
Proof.
intros; eexists (fun x => x).
- econstructor.
+ red; intros; apply LE, IN.
+ intros; apply GF, PR0.
- apply PR.
Qed.
Hint Resolve grespectful5_incl.
Lemma grespectful5_compose: forall clo (RES: respectful5 clo) r,
clo (gres5 r) <5= gres5 r.
Proof.
intros; eapply grespectful5_greatest with (clo := compose clo gres5); [|apply PR].
apply respectful5_compose; [apply RES|apply grespectful5_respectful5].
Qed.
Lemma grespectful5_incl_rev: forall r,
gres5 (paco5 (compose gf gres5) r) <5= paco5 (compose gf gres5) r.
Proof.
intro r; pcofix CIH; intros; pfold.
eapply gf_mon, grespectful5_compose, grespectful5_respectful5.
destruct grespectful5_respectful5; eapply RESPECTFUL0, PR; intros; [apply grespectful5_incl; right; apply CIH, grespectful5_incl, PR0|].
_punfold PR0; [|apply gfgres5_mon].
eapply gfgres5_mon; [apply PR0|].
intros; destruct PR1.
- left. eapply paco5_mon; [apply H| apply CIH0].
- right. eapply CIH0, H.
Qed.
Inductive rclo5 clo (r: rel): rel :=
| rclo5_incl
e0 e1 e2 e3 e4
(R: r e0 e1 e2 e3 e4):
@rclo5 clo r e0 e1 e2 e3 e4
| rclo5_step'
r' e0 e1 e2 e3 e4
(R': r' <5= rclo5 clo r)
(CLOR':clo r' e0 e1 e2 e3 e4):
@rclo5 clo r e0 e1 e2 e3 e4
| rclo5_gf
r' e0 e1 e2 e3 e4
(R': r' <5= rclo5 clo r)
(CLOR':@gf r' e0 e1 e2 e3 e4):
@rclo5 clo r e0 e1 e2 e3 e4
.
Lemma rclo5_mon clo:
monotone5 (rclo5 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 rclo5_mon: paco.
Lemma rclo5_base
clo
(MON: monotone5 clo):
clo <6= rclo5 clo.
Proof.
intros. econstructor 2; [intros; apply PR0|].
eapply MON; [apply PR|intros; constructor; apply PR0].
Qed.
Lemma rclo5_step
(clo: rel -> rel) r:
clo (rclo5 clo r) <5= rclo5 clo r.
Proof.
intros. econstructor 2; [intros; apply PR0|apply PR].
Qed.
Lemma rclo5_rclo5
clo r
(MON: monotone5 clo):
rclo5 clo (rclo5 clo r) <5= rclo5 clo r.
Proof.
intros. induction PR.
- eapply R.
- econstructor 2; [eapply H | eapply CLOR'].
- econstructor 3; [eapply H | eapply CLOR'].
Qed.
Structure weak_respectful5 (clo: rel -> rel) : Prop :=
weak_respectful5_intro {
WEAK_MON: monotone5 clo;
WEAK_RESPECTFUL:
forall l r (LE: l <5= r) (GF: l <5= gf r),
clo l <5= gf (rclo5 clo r);
}.
Hint Constructors weak_respectful5.
Lemma weak_respectful5_respectful5
clo (RES: weak_respectful5 clo):
respectful5 (rclo5 clo).
Proof.
inversion RES. econstructor; [eapply rclo5_mon|]. intros.
induction PR; intros.
- eapply gf_mon; [apply GF, R|]. intros.
apply rclo5_incl. apply PR.
- eapply gf_mon.
+ eapply WEAK_RESPECTFUL0; [|apply H|apply CLOR'].
intros. eapply rclo5_mon; [apply R', PR|apply LE].
+ intros. apply rclo5_rclo5;[apply WEAK_MON0|apply PR].
- eapply gf_mon; [apply CLOR'|].
intros. eapply rclo5_mon; [apply R', PR| apply LE].
Qed.
Lemma upto5_init:
paco5 (compose gf gres5) bot5 <5= paco5 gf bot5.
Proof.
apply sound5_is_gf.
apply respectful5_is_sound5.
apply grespectful5_respectful5.
Qed.
Lemma upto5_final:
paco5 gf <6= paco5 (compose gf gres5).
Proof.
pcofix CIH. intros. _punfold PR; [|apply gf_mon]. pfold.
eapply gf_mon; [|apply grespectful5_incl].
eapply gf_mon; [apply PR|]. intros. right.
inversion PR0; [apply CIH, H | apply CIH0, H].
Qed.
Lemma upto5_step
r clo (RES: weak_respectful5 clo):
clo (paco5 (compose gf gres5) r) <5= paco5 (compose gf gres5) r.
Proof.
intros. apply grespectful5_incl_rev.
assert (RES' := weak_respectful5_respectful5 RES).
eapply grespectful5_greatest; [apply RES'|].
eapply rclo5_base; [apply RES|].
inversion RES. apply PR.
Qed.
Lemma upto5_step_under
r clo (RES: weak_respectful5 clo):
clo (gres5 r) <5= gres5 r.
Proof.
intros. apply weak_respectful5_respectful5 in RES.
eapply grespectful5_compose; [apply RES|].
econstructor 2; [intros; constructor 1; apply PR0 | apply PR].
Qed.
End Respectful5.
Lemma grespectful5_impl T0 T1 T2 T3 T4 (gf gf': rel5 T0 T1 T2 T3 T4 -> rel5 T0 T1 T2 T3 T4) r x0 x1 x2 x3 x4
(PR: gres5 gf r x0 x1 x2 x3 x4)
(EQ: forall r x0 x1 x2 x3 x4, gf r x0 x1 x2 x3 x4 <-> gf' r x0 x1 x2 x3 x4):
gres5 gf' r x0 x1 x2 x3 x4.
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 grespectful5_iff T0 T1 T2 T3 T4 (gf gf': rel5 T0 T1 T2 T3 T4 -> rel5 T0 T1 T2 T3 T4) r x0 x1 x2 x3 x4
(EQ: forall r x0 x1 x2 x3 x4, gf r x0 x1 x2 x3 x4 <-> gf' r x0 x1 x2 x3 x4):
gres5 gf r x0 x1 x2 x3 x4 <-> gres5 gf' r x0 x1 x2 x3 x4.
Proof.
split; intros.
- eapply grespectful5_impl; [apply H | apply EQ].
- eapply grespectful5_impl; [apply H | split; apply EQ].
Qed.
Hint Constructors sound5.
Hint Constructors respectful5.
Hint Constructors gres5.
Hint Resolve gfclo5_mon : paco.
Hint Resolve gfgres5_mon : paco.
Hint Resolve grespectful5_incl.
Hint Resolve rclo5_mon: paco.
Hint Constructors weak_respectful5.
Ltac pupto5_init := eapply upto5_init; [eauto with paco|].
Ltac pupto5_final := first [eapply upto5_final; [eauto with paco|] | eapply grespectful5_incl].
Ltac pupto5 H := first [eapply upto5_step|eapply upto5_step_under]; [|eapply H|]; [eauto with paco|].
Require Import paco5.
Set Implicit Arguments.
Section Respectful5.
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.
Variable T4 : forall (x0: @T0) (x1: @T1 x0) (x2: @T2 x0 x1) (x3: @T3 x0 x1 x2), Type.
Local Notation rel := (rel5 T0 T1 T2 T3 T4).
Variable gf: rel -> rel.
Hypothesis gf_mon: monotone5 gf.
Inductive sound5 (clo: rel -> rel): Prop :=
| sound5_intro
(MON: monotone5 clo)
(SOUND:
forall r (PFIX: r <5= gf (clo r)),
r <5= paco5 gf bot5)
.
Hint Constructors sound5.
Structure respectful5 (clo: rel -> rel) : Prop :=
respectful5_intro {
MON: monotone5 clo;
RESPECTFUL:
forall l r (LE: l <5= r) (GF: l <5= gf r),
clo l <5= gf (clo r);
}.
Hint Constructors respectful5.
Inductive gres5 (r: rel) e0 e1 e2 e3 e4 : Prop :=
| gres5_intro
clo
(RES: respectful5 clo)
(CLO: clo r e0 e1 e2 e3 e4)
.
Hint Constructors gres5.
Lemma gfclo5_mon: forall clo, sound5 clo -> monotone5 (compose gf clo).
Proof.
intros; destruct H; red; intros.
eapply gf_mon; [apply IN|intros; eapply MON0; [apply PR|apply LE]].
Qed.
Hint Resolve gfclo5_mon : paco.
Lemma sound5_is_gf: forall clo (UPTO: sound5 clo),
paco5 (compose gf clo) bot5 <5= paco5 gf bot5.
Proof.
intros. _punfold PR; [|apply gfclo5_mon, UPTO]. edestruct UPTO.
eapply (SOUND (paco5 (compose gf clo) bot5)).
- intros. _punfold PR0; [|apply gfclo5_mon, UPTO].
eapply (gfclo5_mon UPTO); [apply PR0| intros; destruct PR1; [apply H|destruct H]].
- pfold. apply PR.
Qed.
Lemma respectful5_is_sound5: respectful5 <1= sound5.
Proof.
intro clo.
set (rclo := fix rclo clo n (r: rel) :=
match n with
| 0 => r
| S n' => rclo clo n' r \5/ clo (rclo clo n' r)
end).
intros. destruct PR. econstructor; [apply MON0|].
intros. set (rr e0 e1 e2 e3 e4 := exists n, rclo clo n r e0 e1 e2 e3 e4).
assert (rr x0 x1 x2 x3 x4) by (exists 0; apply PR); clear PR.
cut (forall n, rclo clo n r <5= gf (rclo clo (S n) r)).
{ intro X; revert x0 x1 x2 x3 x4 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 respectful5_compose
clo0 clo1
(RES0: respectful5 clo0)
(RES1: respectful5 clo1):
respectful5 (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 grespectful5_mon: monotone5 gres5.
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 grespectful5_respectful5: respectful5 gres5.
Proof.
econstructor; [apply grespectful5_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 gfgres5_mon: monotone5 (compose gf gres5).
Proof.
destruct grespectful5_respectful5.
unfold monotone5. intros. eapply gf_mon; [eapply IN|].
intros. eapply MON0;[apply PR|apply LE].
Qed.
Hint Resolve gfgres5_mon : paco.
Lemma grespectful5_greatest: forall clo (RES: respectful5 clo), clo <6= gres5.
Proof. intros. econstructor;[apply RES|apply PR]. Qed.
Lemma grespectful5_incl: forall r, r <5= gres5 r.
Proof.
intros; eexists (fun x => x).
- econstructor.
+ red; intros; apply LE, IN.
+ intros; apply GF, PR0.
- apply PR.
Qed.
Hint Resolve grespectful5_incl.
Lemma grespectful5_compose: forall clo (RES: respectful5 clo) r,
clo (gres5 r) <5= gres5 r.
Proof.
intros; eapply grespectful5_greatest with (clo := compose clo gres5); [|apply PR].
apply respectful5_compose; [apply RES|apply grespectful5_respectful5].
Qed.
Lemma grespectful5_incl_rev: forall r,
gres5 (paco5 (compose gf gres5) r) <5= paco5 (compose gf gres5) r.
Proof.
intro r; pcofix CIH; intros; pfold.
eapply gf_mon, grespectful5_compose, grespectful5_respectful5.
destruct grespectful5_respectful5; eapply RESPECTFUL0, PR; intros; [apply grespectful5_incl; right; apply CIH, grespectful5_incl, PR0|].
_punfold PR0; [|apply gfgres5_mon].
eapply gfgres5_mon; [apply PR0|].
intros; destruct PR1.
- left. eapply paco5_mon; [apply H| apply CIH0].
- right. eapply CIH0, H.
Qed.
Inductive rclo5 clo (r: rel): rel :=
| rclo5_incl
e0 e1 e2 e3 e4
(R: r e0 e1 e2 e3 e4):
@rclo5 clo r e0 e1 e2 e3 e4
| rclo5_step'
r' e0 e1 e2 e3 e4
(R': r' <5= rclo5 clo r)
(CLOR':clo r' e0 e1 e2 e3 e4):
@rclo5 clo r e0 e1 e2 e3 e4
| rclo5_gf
r' e0 e1 e2 e3 e4
(R': r' <5= rclo5 clo r)
(CLOR':@gf r' e0 e1 e2 e3 e4):
@rclo5 clo r e0 e1 e2 e3 e4
.
Lemma rclo5_mon clo:
monotone5 (rclo5 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 rclo5_mon: paco.
Lemma rclo5_base
clo
(MON: monotone5 clo):
clo <6= rclo5 clo.
Proof.
intros. econstructor 2; [intros; apply PR0|].
eapply MON; [apply PR|intros; constructor; apply PR0].
Qed.
Lemma rclo5_step
(clo: rel -> rel) r:
clo (rclo5 clo r) <5= rclo5 clo r.
Proof.
intros. econstructor 2; [intros; apply PR0|apply PR].
Qed.
Lemma rclo5_rclo5
clo r
(MON: monotone5 clo):
rclo5 clo (rclo5 clo r) <5= rclo5 clo r.
Proof.
intros. induction PR.
- eapply R.
- econstructor 2; [eapply H | eapply CLOR'].
- econstructor 3; [eapply H | eapply CLOR'].
Qed.
Structure weak_respectful5 (clo: rel -> rel) : Prop :=
weak_respectful5_intro {
WEAK_MON: monotone5 clo;
WEAK_RESPECTFUL:
forall l r (LE: l <5= r) (GF: l <5= gf r),
clo l <5= gf (rclo5 clo r);
}.
Hint Constructors weak_respectful5.
Lemma weak_respectful5_respectful5
clo (RES: weak_respectful5 clo):
respectful5 (rclo5 clo).
Proof.
inversion RES. econstructor; [eapply rclo5_mon|]. intros.
induction PR; intros.
- eapply gf_mon; [apply GF, R|]. intros.
apply rclo5_incl. apply PR.
- eapply gf_mon.
+ eapply WEAK_RESPECTFUL0; [|apply H|apply CLOR'].
intros. eapply rclo5_mon; [apply R', PR|apply LE].
+ intros. apply rclo5_rclo5;[apply WEAK_MON0|apply PR].
- eapply gf_mon; [apply CLOR'|].
intros. eapply rclo5_mon; [apply R', PR| apply LE].
Qed.
Lemma upto5_init:
paco5 (compose gf gres5) bot5 <5= paco5 gf bot5.
Proof.
apply sound5_is_gf.
apply respectful5_is_sound5.
apply grespectful5_respectful5.
Qed.
Lemma upto5_final:
paco5 gf <6= paco5 (compose gf gres5).
Proof.
pcofix CIH. intros. _punfold PR; [|apply gf_mon]. pfold.
eapply gf_mon; [|apply grespectful5_incl].
eapply gf_mon; [apply PR|]. intros. right.
inversion PR0; [apply CIH, H | apply CIH0, H].
Qed.
Lemma upto5_step
r clo (RES: weak_respectful5 clo):
clo (paco5 (compose gf gres5) r) <5= paco5 (compose gf gres5) r.
Proof.
intros. apply grespectful5_incl_rev.
assert (RES' := weak_respectful5_respectful5 RES).
eapply grespectful5_greatest; [apply RES'|].
eapply rclo5_base; [apply RES|].
inversion RES. apply PR.
Qed.
Lemma upto5_step_under
r clo (RES: weak_respectful5 clo):
clo (gres5 r) <5= gres5 r.
Proof.
intros. apply weak_respectful5_respectful5 in RES.
eapply grespectful5_compose; [apply RES|].
econstructor 2; [intros; constructor 1; apply PR0 | apply PR].
Qed.
End Respectful5.
Lemma grespectful5_impl T0 T1 T2 T3 T4 (gf gf': rel5 T0 T1 T2 T3 T4 -> rel5 T0 T1 T2 T3 T4) r x0 x1 x2 x3 x4
(PR: gres5 gf r x0 x1 x2 x3 x4)
(EQ: forall r x0 x1 x2 x3 x4, gf r x0 x1 x2 x3 x4 <-> gf' r x0 x1 x2 x3 x4):
gres5 gf' r x0 x1 x2 x3 x4.
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 grespectful5_iff T0 T1 T2 T3 T4 (gf gf': rel5 T0 T1 T2 T3 T4 -> rel5 T0 T1 T2 T3 T4) r x0 x1 x2 x3 x4
(EQ: forall r x0 x1 x2 x3 x4, gf r x0 x1 x2 x3 x4 <-> gf' r x0 x1 x2 x3 x4):
gres5 gf r x0 x1 x2 x3 x4 <-> gres5 gf' r x0 x1 x2 x3 x4.
Proof.
split; intros.
- eapply grespectful5_impl; [apply H | apply EQ].
- eapply grespectful5_impl; [apply H | split; apply EQ].
Qed.
Hint Constructors sound5.
Hint Constructors respectful5.
Hint Constructors gres5.
Hint Resolve gfclo5_mon : paco.
Hint Resolve gfgres5_mon : paco.
Hint Resolve grespectful5_incl.
Hint Resolve rclo5_mon: paco.
Hint Constructors weak_respectful5.
Ltac pupto5_init := eapply upto5_init; [eauto with paco|].
Ltac pupto5_final := first [eapply upto5_final; [eauto with paco|] | eapply grespectful5_incl].
Ltac pupto5 H := first [eapply upto5_step|eapply upto5_step_under]; [|eapply H|]; [eauto with paco|].