Library Paco.paco8_upto
Require Export Program.Basics. Open Scope program_scope.
Require Import paco8.
Set Implicit Arguments.
Section Respectful8.
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.
Variable T5 : forall (x0: @T0) (x1: @T1 x0) (x2: @T2 x0 x1) (x3: @T3 x0 x1 x2) (x4: @T4 x0 x1 x2 x3), Type.
Variable T6 : forall (x0: @T0) (x1: @T1 x0) (x2: @T2 x0 x1) (x3: @T3 x0 x1 x2) (x4: @T4 x0 x1 x2 x3) (x5: @T5 x0 x1 x2 x3 x4), Type.
Variable T7 : forall (x0: @T0) (x1: @T1 x0) (x2: @T2 x0 x1) (x3: @T3 x0 x1 x2) (x4: @T4 x0 x1 x2 x3) (x5: @T5 x0 x1 x2 x3 x4) (x6: @T6 x0 x1 x2 x3 x4 x5), Type.
Local Notation rel := (rel8 T0 T1 T2 T3 T4 T5 T6 T7).
Variable gf: rel -> rel.
Hypothesis gf_mon: monotone8 gf.
Inductive sound8 (clo: rel -> rel): Prop :=
| sound8_intro
(MON: monotone8 clo)
(SOUND:
forall r (PFIX: r <8= gf (clo r)),
r <8= paco8 gf bot8)
.
Hint Constructors sound8.
Structure respectful8 (clo: rel -> rel) : Prop :=
respectful8_intro {
MON: monotone8 clo;
RESPECTFUL:
forall l r (LE: l <8= r) (GF: l <8= gf r),
clo l <8= gf (clo r);
}.
Hint Constructors respectful8.
Inductive gres8 (r: rel) e0 e1 e2 e3 e4 e5 e6 e7 : Prop :=
| gres8_intro
clo
(RES: respectful8 clo)
(CLO: clo r e0 e1 e2 e3 e4 e5 e6 e7)
.
Hint Constructors gres8.
Lemma gfclo8_mon: forall clo, sound8 clo -> monotone8 (compose gf clo).
Proof.
intros; destruct H; red; intros.
eapply gf_mon; [apply IN|intros; eapply MON0; [apply PR|apply LE]].
Qed.
Hint Resolve gfclo8_mon : paco.
Lemma sound8_is_gf: forall clo (UPTO: sound8 clo),
paco8 (compose gf clo) bot8 <8= paco8 gf bot8.
Proof.
intros. _punfold PR; [|apply gfclo8_mon, UPTO]. edestruct UPTO.
eapply (SOUND (paco8 (compose gf clo) bot8)).
- intros. _punfold PR0; [|apply gfclo8_mon, UPTO].
eapply (gfclo8_mon UPTO); [apply PR0| intros; destruct PR1; [apply H|destruct H]].
- pfold. apply PR.
Qed.
Lemma respectful8_is_sound8: respectful8 <1= sound8.
Proof.
intro clo.
set (rclo := fix rclo clo n (r: rel) :=
match n with
| 0 => r
| S n' => rclo clo n' r \8/ clo (rclo clo n' r)
end).
intros. destruct PR. econstructor; [apply MON0|].
intros. set (rr e0 e1 e2 e3 e4 e5 e6 e7 := exists n, rclo clo n r e0 e1 e2 e3 e4 e5 e6 e7).
assert (rr x0 x1 x2 x3 x4 x5 x6 x7) by (exists 0; apply PR); clear PR.
cut (forall n, rclo clo n r <8= gf (rclo clo (S n) r)).
{ intro X; revert x0 x1 x2 x3 x4 x5 x6 x7 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 respectful8_compose
clo0 clo1
(RES0: respectful8 clo0)
(RES1: respectful8 clo1):
respectful8 (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 grespectful8_mon: monotone8 gres8.
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 grespectful8_respectful8: respectful8 gres8.
Proof.
econstructor; [apply grespectful8_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 gfgres8_mon: monotone8 (compose gf gres8).
Proof.
destruct grespectful8_respectful8.
unfold monotone8. intros. eapply gf_mon; [eapply IN|].
intros. eapply MON0;[apply PR|apply LE].
Qed.
Hint Resolve gfgres8_mon : paco.
Lemma grespectful8_greatest: forall clo (RES: respectful8 clo), clo <9= gres8.
Proof. intros. econstructor;[apply RES|apply PR]. Qed.
Lemma grespectful8_incl: forall r, r <8= gres8 r.
Proof.
intros; eexists (fun x => x).
- econstructor.
+ red; intros; apply LE, IN.
+ intros; apply GF, PR0.
- apply PR.
Qed.
Hint Resolve grespectful8_incl.
Lemma grespectful8_compose: forall clo (RES: respectful8 clo) r,
clo (gres8 r) <8= gres8 r.
Proof.
intros; eapply grespectful8_greatest with (clo := compose clo gres8); [|apply PR].
apply respectful8_compose; [apply RES|apply grespectful8_respectful8].
Qed.
Lemma grespectful8_incl_rev: forall r,
gres8 (paco8 (compose gf gres8) r) <8= paco8 (compose gf gres8) r.
Proof.
intro r; pcofix CIH; intros; pfold.
eapply gf_mon, grespectful8_compose, grespectful8_respectful8.
destruct grespectful8_respectful8; eapply RESPECTFUL0, PR; intros; [apply grespectful8_incl; right; apply CIH, grespectful8_incl, PR0|].
_punfold PR0; [|apply gfgres8_mon].
eapply gfgres8_mon; [apply PR0|].
intros; destruct PR1.
- left. eapply paco8_mon; [apply H| apply CIH0].
- right. eapply CIH0, H.
Qed.
Inductive rclo8 clo (r: rel): rel :=
| rclo8_incl
e0 e1 e2 e3 e4 e5 e6 e7
(R: r e0 e1 e2 e3 e4 e5 e6 e7):
@rclo8 clo r e0 e1 e2 e3 e4 e5 e6 e7
| rclo8_step'
r' e0 e1 e2 e3 e4 e5 e6 e7
(R': r' <8= rclo8 clo r)
(CLOR':clo r' e0 e1 e2 e3 e4 e5 e6 e7):
@rclo8 clo r e0 e1 e2 e3 e4 e5 e6 e7
| rclo8_gf
r' e0 e1 e2 e3 e4 e5 e6 e7
(R': r' <8= rclo8 clo r)
(CLOR':@gf r' e0 e1 e2 e3 e4 e5 e6 e7):
@rclo8 clo r e0 e1 e2 e3 e4 e5 e6 e7
.
Lemma rclo8_mon clo:
monotone8 (rclo8 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 rclo8_mon: paco.
Lemma rclo8_base
clo
(MON: monotone8 clo):
clo <9= rclo8 clo.
Proof.
intros. econstructor 2; [intros; apply PR0|].
eapply MON; [apply PR|intros; constructor; apply PR0].
Qed.
Lemma rclo8_step
(clo: rel -> rel) r:
clo (rclo8 clo r) <8= rclo8 clo r.
Proof.
intros. econstructor 2; [intros; apply PR0|apply PR].
Qed.
Lemma rclo8_rclo8
clo r
(MON: monotone8 clo):
rclo8 clo (rclo8 clo r) <8= rclo8 clo r.
Proof.
intros. induction PR.
- eapply R.
- econstructor 2; [eapply H | eapply CLOR'].
- econstructor 3; [eapply H | eapply CLOR'].
Qed.
Structure weak_respectful8 (clo: rel -> rel) : Prop :=
weak_respectful8_intro {
WEAK_MON: monotone8 clo;
WEAK_RESPECTFUL:
forall l r (LE: l <8= r) (GF: l <8= gf r),
clo l <8= gf (rclo8 clo r);
}.
Hint Constructors weak_respectful8.
Lemma weak_respectful8_respectful8
clo (RES: weak_respectful8 clo):
respectful8 (rclo8 clo).
Proof.
inversion RES. econstructor; [eapply rclo8_mon|]. intros.
induction PR; intros.
- eapply gf_mon; [apply GF, R|]. intros.
apply rclo8_incl. apply PR.
- eapply gf_mon.
+ eapply WEAK_RESPECTFUL0; [|apply H|apply CLOR'].
intros. eapply rclo8_mon; [apply R', PR|apply LE].
+ intros. apply rclo8_rclo8;[apply WEAK_MON0|apply PR].
- eapply gf_mon; [apply CLOR'|].
intros. eapply rclo8_mon; [apply R', PR| apply LE].
Qed.
Lemma upto8_init:
paco8 (compose gf gres8) bot8 <8= paco8 gf bot8.
Proof.
apply sound8_is_gf.
apply respectful8_is_sound8.
apply grespectful8_respectful8.
Qed.
Lemma upto8_final:
paco8 gf <9= paco8 (compose gf gres8).
Proof.
pcofix CIH. intros. _punfold PR; [|apply gf_mon]. pfold.
eapply gf_mon; [|apply grespectful8_incl].
eapply gf_mon; [apply PR|]. intros. right.
inversion PR0; [apply CIH, H | apply CIH0, H].
Qed.
Lemma upto8_step
r clo (RES: weak_respectful8 clo):
clo (paco8 (compose gf gres8) r) <8= paco8 (compose gf gres8) r.
Proof.
intros. apply grespectful8_incl_rev.
assert (RES' := weak_respectful8_respectful8 RES).
eapply grespectful8_greatest; [apply RES'|].
eapply rclo8_base; [apply RES|].
inversion RES. apply PR.
Qed.
Lemma upto8_step_under
r clo (RES: weak_respectful8 clo):
clo (gres8 r) <8= gres8 r.
Proof.
intros. apply weak_respectful8_respectful8 in RES.
eapply grespectful8_compose; [apply RES|].
econstructor 2; [intros; constructor 1; apply PR0 | apply PR].
Qed.
End Respectful8.
Lemma grespectful8_impl T0 T1 T2 T3 T4 T5 T6 T7 (gf gf': rel8 T0 T1 T2 T3 T4 T5 T6 T7 -> rel8 T0 T1 T2 T3 T4 T5 T6 T7) r x0 x1 x2 x3 x4 x5 x6 x7
(PR: gres8 gf r x0 x1 x2 x3 x4 x5 x6 x7)
(EQ: forall r x0 x1 x2 x3 x4 x5 x6 x7, gf r x0 x1 x2 x3 x4 x5 x6 x7 <-> gf' r x0 x1 x2 x3 x4 x5 x6 x7):
gres8 gf' r x0 x1 x2 x3 x4 x5 x6 x7.
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 grespectful8_iff T0 T1 T2 T3 T4 T5 T6 T7 (gf gf': rel8 T0 T1 T2 T3 T4 T5 T6 T7 -> rel8 T0 T1 T2 T3 T4 T5 T6 T7) r x0 x1 x2 x3 x4 x5 x6 x7
(EQ: forall r x0 x1 x2 x3 x4 x5 x6 x7, gf r x0 x1 x2 x3 x4 x5 x6 x7 <-> gf' r x0 x1 x2 x3 x4 x5 x6 x7):
gres8 gf r x0 x1 x2 x3 x4 x5 x6 x7 <-> gres8 gf' r x0 x1 x2 x3 x4 x5 x6 x7.
Proof.
split; intros.
- eapply grespectful8_impl; [apply H | apply EQ].
- eapply grespectful8_impl; [apply H | split; apply EQ].
Qed.
Hint Constructors sound8.
Hint Constructors respectful8.
Hint Constructors gres8.
Hint Resolve gfclo8_mon : paco.
Hint Resolve gfgres8_mon : paco.
Hint Resolve grespectful8_incl.
Hint Resolve rclo8_mon: paco.
Hint Constructors weak_respectful8.
Ltac pupto8_init := eapply upto8_init; [eauto with paco|].
Ltac pupto8_final := first [eapply upto8_final; [eauto with paco|] | eapply grespectful8_incl].
Ltac pupto8 H := first [eapply upto8_step|eapply upto8_step_under]; [|eapply H|]; [eauto with paco|].
Require Import paco8.
Set Implicit Arguments.
Section Respectful8.
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.
Variable T5 : forall (x0: @T0) (x1: @T1 x0) (x2: @T2 x0 x1) (x3: @T3 x0 x1 x2) (x4: @T4 x0 x1 x2 x3), Type.
Variable T6 : forall (x0: @T0) (x1: @T1 x0) (x2: @T2 x0 x1) (x3: @T3 x0 x1 x2) (x4: @T4 x0 x1 x2 x3) (x5: @T5 x0 x1 x2 x3 x4), Type.
Variable T7 : forall (x0: @T0) (x1: @T1 x0) (x2: @T2 x0 x1) (x3: @T3 x0 x1 x2) (x4: @T4 x0 x1 x2 x3) (x5: @T5 x0 x1 x2 x3 x4) (x6: @T6 x0 x1 x2 x3 x4 x5), Type.
Local Notation rel := (rel8 T0 T1 T2 T3 T4 T5 T6 T7).
Variable gf: rel -> rel.
Hypothesis gf_mon: monotone8 gf.
Inductive sound8 (clo: rel -> rel): Prop :=
| sound8_intro
(MON: monotone8 clo)
(SOUND:
forall r (PFIX: r <8= gf (clo r)),
r <8= paco8 gf bot8)
.
Hint Constructors sound8.
Structure respectful8 (clo: rel -> rel) : Prop :=
respectful8_intro {
MON: monotone8 clo;
RESPECTFUL:
forall l r (LE: l <8= r) (GF: l <8= gf r),
clo l <8= gf (clo r);
}.
Hint Constructors respectful8.
Inductive gres8 (r: rel) e0 e1 e2 e3 e4 e5 e6 e7 : Prop :=
| gres8_intro
clo
(RES: respectful8 clo)
(CLO: clo r e0 e1 e2 e3 e4 e5 e6 e7)
.
Hint Constructors gres8.
Lemma gfclo8_mon: forall clo, sound8 clo -> monotone8 (compose gf clo).
Proof.
intros; destruct H; red; intros.
eapply gf_mon; [apply IN|intros; eapply MON0; [apply PR|apply LE]].
Qed.
Hint Resolve gfclo8_mon : paco.
Lemma sound8_is_gf: forall clo (UPTO: sound8 clo),
paco8 (compose gf clo) bot8 <8= paco8 gf bot8.
Proof.
intros. _punfold PR; [|apply gfclo8_mon, UPTO]. edestruct UPTO.
eapply (SOUND (paco8 (compose gf clo) bot8)).
- intros. _punfold PR0; [|apply gfclo8_mon, UPTO].
eapply (gfclo8_mon UPTO); [apply PR0| intros; destruct PR1; [apply H|destruct H]].
- pfold. apply PR.
Qed.
Lemma respectful8_is_sound8: respectful8 <1= sound8.
Proof.
intro clo.
set (rclo := fix rclo clo n (r: rel) :=
match n with
| 0 => r
| S n' => rclo clo n' r \8/ clo (rclo clo n' r)
end).
intros. destruct PR. econstructor; [apply MON0|].
intros. set (rr e0 e1 e2 e3 e4 e5 e6 e7 := exists n, rclo clo n r e0 e1 e2 e3 e4 e5 e6 e7).
assert (rr x0 x1 x2 x3 x4 x5 x6 x7) by (exists 0; apply PR); clear PR.
cut (forall n, rclo clo n r <8= gf (rclo clo (S n) r)).
{ intro X; revert x0 x1 x2 x3 x4 x5 x6 x7 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 respectful8_compose
clo0 clo1
(RES0: respectful8 clo0)
(RES1: respectful8 clo1):
respectful8 (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 grespectful8_mon: monotone8 gres8.
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 grespectful8_respectful8: respectful8 gres8.
Proof.
econstructor; [apply grespectful8_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 gfgres8_mon: monotone8 (compose gf gres8).
Proof.
destruct grespectful8_respectful8.
unfold monotone8. intros. eapply gf_mon; [eapply IN|].
intros. eapply MON0;[apply PR|apply LE].
Qed.
Hint Resolve gfgres8_mon : paco.
Lemma grespectful8_greatest: forall clo (RES: respectful8 clo), clo <9= gres8.
Proof. intros. econstructor;[apply RES|apply PR]. Qed.
Lemma grespectful8_incl: forall r, r <8= gres8 r.
Proof.
intros; eexists (fun x => x).
- econstructor.
+ red; intros; apply LE, IN.
+ intros; apply GF, PR0.
- apply PR.
Qed.
Hint Resolve grespectful8_incl.
Lemma grespectful8_compose: forall clo (RES: respectful8 clo) r,
clo (gres8 r) <8= gres8 r.
Proof.
intros; eapply grespectful8_greatest with (clo := compose clo gres8); [|apply PR].
apply respectful8_compose; [apply RES|apply grespectful8_respectful8].
Qed.
Lemma grespectful8_incl_rev: forall r,
gres8 (paco8 (compose gf gres8) r) <8= paco8 (compose gf gres8) r.
Proof.
intro r; pcofix CIH; intros; pfold.
eapply gf_mon, grespectful8_compose, grespectful8_respectful8.
destruct grespectful8_respectful8; eapply RESPECTFUL0, PR; intros; [apply grespectful8_incl; right; apply CIH, grespectful8_incl, PR0|].
_punfold PR0; [|apply gfgres8_mon].
eapply gfgres8_mon; [apply PR0|].
intros; destruct PR1.
- left. eapply paco8_mon; [apply H| apply CIH0].
- right. eapply CIH0, H.
Qed.
Inductive rclo8 clo (r: rel): rel :=
| rclo8_incl
e0 e1 e2 e3 e4 e5 e6 e7
(R: r e0 e1 e2 e3 e4 e5 e6 e7):
@rclo8 clo r e0 e1 e2 e3 e4 e5 e6 e7
| rclo8_step'
r' e0 e1 e2 e3 e4 e5 e6 e7
(R': r' <8= rclo8 clo r)
(CLOR':clo r' e0 e1 e2 e3 e4 e5 e6 e7):
@rclo8 clo r e0 e1 e2 e3 e4 e5 e6 e7
| rclo8_gf
r' e0 e1 e2 e3 e4 e5 e6 e7
(R': r' <8= rclo8 clo r)
(CLOR':@gf r' e0 e1 e2 e3 e4 e5 e6 e7):
@rclo8 clo r e0 e1 e2 e3 e4 e5 e6 e7
.
Lemma rclo8_mon clo:
monotone8 (rclo8 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 rclo8_mon: paco.
Lemma rclo8_base
clo
(MON: monotone8 clo):
clo <9= rclo8 clo.
Proof.
intros. econstructor 2; [intros; apply PR0|].
eapply MON; [apply PR|intros; constructor; apply PR0].
Qed.
Lemma rclo8_step
(clo: rel -> rel) r:
clo (rclo8 clo r) <8= rclo8 clo r.
Proof.
intros. econstructor 2; [intros; apply PR0|apply PR].
Qed.
Lemma rclo8_rclo8
clo r
(MON: monotone8 clo):
rclo8 clo (rclo8 clo r) <8= rclo8 clo r.
Proof.
intros. induction PR.
- eapply R.
- econstructor 2; [eapply H | eapply CLOR'].
- econstructor 3; [eapply H | eapply CLOR'].
Qed.
Structure weak_respectful8 (clo: rel -> rel) : Prop :=
weak_respectful8_intro {
WEAK_MON: monotone8 clo;
WEAK_RESPECTFUL:
forall l r (LE: l <8= r) (GF: l <8= gf r),
clo l <8= gf (rclo8 clo r);
}.
Hint Constructors weak_respectful8.
Lemma weak_respectful8_respectful8
clo (RES: weak_respectful8 clo):
respectful8 (rclo8 clo).
Proof.
inversion RES. econstructor; [eapply rclo8_mon|]. intros.
induction PR; intros.
- eapply gf_mon; [apply GF, R|]. intros.
apply rclo8_incl. apply PR.
- eapply gf_mon.
+ eapply WEAK_RESPECTFUL0; [|apply H|apply CLOR'].
intros. eapply rclo8_mon; [apply R', PR|apply LE].
+ intros. apply rclo8_rclo8;[apply WEAK_MON0|apply PR].
- eapply gf_mon; [apply CLOR'|].
intros. eapply rclo8_mon; [apply R', PR| apply LE].
Qed.
Lemma upto8_init:
paco8 (compose gf gres8) bot8 <8= paco8 gf bot8.
Proof.
apply sound8_is_gf.
apply respectful8_is_sound8.
apply grespectful8_respectful8.
Qed.
Lemma upto8_final:
paco8 gf <9= paco8 (compose gf gres8).
Proof.
pcofix CIH. intros. _punfold PR; [|apply gf_mon]. pfold.
eapply gf_mon; [|apply grespectful8_incl].
eapply gf_mon; [apply PR|]. intros. right.
inversion PR0; [apply CIH, H | apply CIH0, H].
Qed.
Lemma upto8_step
r clo (RES: weak_respectful8 clo):
clo (paco8 (compose gf gres8) r) <8= paco8 (compose gf gres8) r.
Proof.
intros. apply grespectful8_incl_rev.
assert (RES' := weak_respectful8_respectful8 RES).
eapply grespectful8_greatest; [apply RES'|].
eapply rclo8_base; [apply RES|].
inversion RES. apply PR.
Qed.
Lemma upto8_step_under
r clo (RES: weak_respectful8 clo):
clo (gres8 r) <8= gres8 r.
Proof.
intros. apply weak_respectful8_respectful8 in RES.
eapply grespectful8_compose; [apply RES|].
econstructor 2; [intros; constructor 1; apply PR0 | apply PR].
Qed.
End Respectful8.
Lemma grespectful8_impl T0 T1 T2 T3 T4 T5 T6 T7 (gf gf': rel8 T0 T1 T2 T3 T4 T5 T6 T7 -> rel8 T0 T1 T2 T3 T4 T5 T6 T7) r x0 x1 x2 x3 x4 x5 x6 x7
(PR: gres8 gf r x0 x1 x2 x3 x4 x5 x6 x7)
(EQ: forall r x0 x1 x2 x3 x4 x5 x6 x7, gf r x0 x1 x2 x3 x4 x5 x6 x7 <-> gf' r x0 x1 x2 x3 x4 x5 x6 x7):
gres8 gf' r x0 x1 x2 x3 x4 x5 x6 x7.
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 grespectful8_iff T0 T1 T2 T3 T4 T5 T6 T7 (gf gf': rel8 T0 T1 T2 T3 T4 T5 T6 T7 -> rel8 T0 T1 T2 T3 T4 T5 T6 T7) r x0 x1 x2 x3 x4 x5 x6 x7
(EQ: forall r x0 x1 x2 x3 x4 x5 x6 x7, gf r x0 x1 x2 x3 x4 x5 x6 x7 <-> gf' r x0 x1 x2 x3 x4 x5 x6 x7):
gres8 gf r x0 x1 x2 x3 x4 x5 x6 x7 <-> gres8 gf' r x0 x1 x2 x3 x4 x5 x6 x7.
Proof.
split; intros.
- eapply grespectful8_impl; [apply H | apply EQ].
- eapply grespectful8_impl; [apply H | split; apply EQ].
Qed.
Hint Constructors sound8.
Hint Constructors respectful8.
Hint Constructors gres8.
Hint Resolve gfclo8_mon : paco.
Hint Resolve gfgres8_mon : paco.
Hint Resolve grespectful8_incl.
Hint Resolve rclo8_mon: paco.
Hint Constructors weak_respectful8.
Ltac pupto8_init := eapply upto8_init; [eauto with paco|].
Ltac pupto8_final := first [eapply upto8_final; [eauto with paco|] | eapply grespectful8_incl].
Ltac pupto8 H := first [eapply upto8_step|eapply upto8_step_under]; [|eapply H|]; [eauto with paco|].