Library Paco.paco12_upto
Require Export Program.Basics. Open Scope program_scope.
Require Import paco12.
Set Implicit Arguments.
Section Respectful12.
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.
Variable T8 : 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) (x7: @T7 x0 x1 x2 x3 x4 x5 x6), Type.
Variable T9 : 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) (x7: @T7 x0 x1 x2 x3 x4 x5 x6) (x8: @T8 x0 x1 x2 x3 x4 x5 x6 x7), Type.
Variable T10 : 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) (x7: @T7 x0 x1 x2 x3 x4 x5 x6) (x8: @T8 x0 x1 x2 x3 x4 x5 x6 x7) (x9: @T9 x0 x1 x2 x3 x4 x5 x6 x7 x8), Type.
Variable T11 : 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) (x7: @T7 x0 x1 x2 x3 x4 x5 x6) (x8: @T8 x0 x1 x2 x3 x4 x5 x6 x7) (x9: @T9 x0 x1 x2 x3 x4 x5 x6 x7 x8) (x10: @T10 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9), Type.
Local Notation rel := (rel12 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11).
Variable gf: rel -> rel.
Hypothesis gf_mon: monotone12 gf.
Inductive sound12 (clo: rel -> rel): Prop :=
| sound12_intro
(MON: monotone12 clo)
(SOUND:
forall r (PFIX: r <12= gf (clo r)),
r <12= paco12 gf bot12)
.
Hint Constructors sound12.
Structure respectful12 (clo: rel -> rel) : Prop :=
respectful12_intro {
MON: monotone12 clo;
RESPECTFUL:
forall l r (LE: l <12= r) (GF: l <12= gf r),
clo l <12= gf (clo r);
}.
Hint Constructors respectful12.
Inductive gres12 (r: rel) e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 : Prop :=
| gres12_intro
clo
(RES: respectful12 clo)
(CLO: clo r e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11)
.
Hint Constructors gres12.
Lemma gfclo12_mon: forall clo, sound12 clo -> monotone12 (compose gf clo).
Proof.
intros; destruct H; red; intros.
eapply gf_mon; [apply IN|intros; eapply MON0; [apply PR|apply LE]].
Qed.
Hint Resolve gfclo12_mon : paco.
Lemma sound12_is_gf: forall clo (UPTO: sound12 clo),
paco12 (compose gf clo) bot12 <12= paco12 gf bot12.
Proof.
intros. _punfold PR; [|apply gfclo12_mon, UPTO]. edestruct UPTO.
eapply (SOUND (paco12 (compose gf clo) bot12)).
- intros. _punfold PR0; [|apply gfclo12_mon, UPTO].
eapply (gfclo12_mon UPTO); [apply PR0| intros; destruct PR1; [apply H|destruct H]].
- pfold. apply PR.
Qed.
Lemma respectful12_is_sound12: respectful12 <1= sound12.
Proof.
intro clo.
set (rclo := fix rclo clo n (r: rel) :=
match n with
| 0 => r
| S n' => rclo clo n' r \12/ clo (rclo clo n' r)
end).
intros. destruct PR. econstructor; [apply MON0|].
intros. set (rr e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 := exists n, rclo clo n r e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11).
assert (rr x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11) by (exists 0; apply PR); clear PR.
cut (forall n, rclo clo n r <12= gf (rclo clo (S n) r)).
{ intro X; revert x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 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 respectful12_compose
clo0 clo1
(RES0: respectful12 clo0)
(RES1: respectful12 clo1):
respectful12 (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 grespectful12_mon: monotone12 gres12.
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 grespectful12_respectful12: respectful12 gres12.
Proof.
econstructor; [apply grespectful12_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 gfgres12_mon: monotone12 (compose gf gres12).
Proof.
destruct grespectful12_respectful12.
unfold monotone12. intros. eapply gf_mon; [eapply IN|].
intros. eapply MON0;[apply PR|apply LE].
Qed.
Hint Resolve gfgres12_mon : paco.
Lemma grespectful12_greatest: forall clo (RES: respectful12 clo), clo <13= gres12.
Proof. intros. econstructor;[apply RES|apply PR]. Qed.
Lemma grespectful12_incl: forall r, r <12= gres12 r.
Proof.
intros; eexists (fun x => x).
- econstructor.
+ red; intros; apply LE, IN.
+ intros; apply GF, PR0.
- apply PR.
Qed.
Hint Resolve grespectful12_incl.
Lemma grespectful12_compose: forall clo (RES: respectful12 clo) r,
clo (gres12 r) <12= gres12 r.
Proof.
intros; eapply grespectful12_greatest with (clo := compose clo gres12); [|apply PR].
apply respectful12_compose; [apply RES|apply grespectful12_respectful12].
Qed.
Lemma grespectful12_incl_rev: forall r,
gres12 (paco12 (compose gf gres12) r) <12= paco12 (compose gf gres12) r.
Proof.
intro r; pcofix CIH; intros; pfold.
eapply gf_mon, grespectful12_compose, grespectful12_respectful12.
destruct grespectful12_respectful12; eapply RESPECTFUL0, PR; intros; [apply grespectful12_incl; right; apply CIH, grespectful12_incl, PR0|].
_punfold PR0; [|apply gfgres12_mon].
eapply gfgres12_mon; [apply PR0|].
intros; destruct PR1.
- left. eapply paco12_mon; [apply H| apply CIH0].
- right. eapply CIH0, H.
Qed.
Inductive rclo12 clo (r: rel): rel :=
| rclo12_incl
e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11
(R: r e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11):
@rclo12 clo r e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11
| rclo12_step'
r' e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11
(R': r' <12= rclo12 clo r)
(CLOR':clo r' e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11):
@rclo12 clo r e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11
| rclo12_gf
r' e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11
(R': r' <12= rclo12 clo r)
(CLOR':@gf r' e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11):
@rclo12 clo r e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11
.
Lemma rclo12_mon clo:
monotone12 (rclo12 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 rclo12_mon: paco.
Lemma rclo12_base
clo
(MON: monotone12 clo):
clo <13= rclo12 clo.
Proof.
intros. econstructor 2; [intros; apply PR0|].
eapply MON; [apply PR|intros; constructor; apply PR0].
Qed.
Lemma rclo12_step
(clo: rel -> rel) r:
clo (rclo12 clo r) <12= rclo12 clo r.
Proof.
intros. econstructor 2; [intros; apply PR0|apply PR].
Qed.
Lemma rclo12_rclo12
clo r
(MON: monotone12 clo):
rclo12 clo (rclo12 clo r) <12= rclo12 clo r.
Proof.
intros. induction PR.
- eapply R.
- econstructor 2; [eapply H | eapply CLOR'].
- econstructor 3; [eapply H | eapply CLOR'].
Qed.
Structure weak_respectful12 (clo: rel -> rel) : Prop :=
weak_respectful12_intro {
WEAK_MON: monotone12 clo;
WEAK_RESPECTFUL:
forall l r (LE: l <12= r) (GF: l <12= gf r),
clo l <12= gf (rclo12 clo r);
}.
Hint Constructors weak_respectful12.
Lemma weak_respectful12_respectful12
clo (RES: weak_respectful12 clo):
respectful12 (rclo12 clo).
Proof.
inversion RES. econstructor; [eapply rclo12_mon|]. intros.
induction PR; intros.
- eapply gf_mon; [apply GF, R|]. intros.
apply rclo12_incl. apply PR.
- eapply gf_mon.
+ eapply WEAK_RESPECTFUL0; [|apply H|apply CLOR'].
intros. eapply rclo12_mon; [apply R', PR|apply LE].
+ intros. apply rclo12_rclo12;[apply WEAK_MON0|apply PR].
- eapply gf_mon; [apply CLOR'|].
intros. eapply rclo12_mon; [apply R', PR| apply LE].
Qed.
Lemma upto12_init:
paco12 (compose gf gres12) bot12 <12= paco12 gf bot12.
Proof.
apply sound12_is_gf.
apply respectful12_is_sound12.
apply grespectful12_respectful12.
Qed.
Lemma upto12_final:
paco12 gf <13= paco12 (compose gf gres12).
Proof.
pcofix CIH. intros. _punfold PR; [|apply gf_mon]. pfold.
eapply gf_mon; [|apply grespectful12_incl].
eapply gf_mon; [apply PR|]. intros. right.
inversion PR0; [apply CIH, H | apply CIH0, H].
Qed.
Lemma upto12_step
r clo (RES: weak_respectful12 clo):
clo (paco12 (compose gf gres12) r) <12= paco12 (compose gf gres12) r.
Proof.
intros. apply grespectful12_incl_rev.
assert (RES' := weak_respectful12_respectful12 RES).
eapply grespectful12_greatest; [apply RES'|].
eapply rclo12_base; [apply RES|].
inversion RES. apply PR.
Qed.
Lemma upto12_step_under
r clo (RES: weak_respectful12 clo):
clo (gres12 r) <12= gres12 r.
Proof.
intros. apply weak_respectful12_respectful12 in RES.
eapply grespectful12_compose; [apply RES|].
econstructor 2; [intros; constructor 1; apply PR0 | apply PR].
Qed.
End Respectful12.
Lemma grespectful12_impl T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 (gf gf': rel12 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 -> rel12 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11) r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11
(PR: gres12 gf r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11)
(EQ: forall r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11, gf r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 <-> gf' r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11):
gres12 gf' r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11.
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 grespectful12_iff T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 (gf gf': rel12 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 -> rel12 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11) r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11
(EQ: forall r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11, gf r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 <-> gf' r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11):
gres12 gf r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 <-> gres12 gf' r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11.
Proof.
split; intros.
- eapply grespectful12_impl; [apply H | apply EQ].
- eapply grespectful12_impl; [apply H | split; apply EQ].
Qed.
Hint Constructors sound12.
Hint Constructors respectful12.
Hint Constructors gres12.
Hint Resolve gfclo12_mon : paco.
Hint Resolve gfgres12_mon : paco.
Hint Resolve grespectful12_incl.
Hint Resolve rclo12_mon: paco.
Hint Constructors weak_respectful12.
Ltac pupto12_init := eapply upto12_init; [eauto with paco|].
Ltac pupto12_final := first [eapply upto12_final; [eauto with paco|] | eapply grespectful12_incl].
Ltac pupto12 H := first [eapply upto12_step|eapply upto12_step_under]; [|eapply H|]; [eauto with paco|].
Require Import paco12.
Set Implicit Arguments.
Section Respectful12.
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.
Variable T8 : 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) (x7: @T7 x0 x1 x2 x3 x4 x5 x6), Type.
Variable T9 : 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) (x7: @T7 x0 x1 x2 x3 x4 x5 x6) (x8: @T8 x0 x1 x2 x3 x4 x5 x6 x7), Type.
Variable T10 : 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) (x7: @T7 x0 x1 x2 x3 x4 x5 x6) (x8: @T8 x0 x1 x2 x3 x4 x5 x6 x7) (x9: @T9 x0 x1 x2 x3 x4 x5 x6 x7 x8), Type.
Variable T11 : 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) (x7: @T7 x0 x1 x2 x3 x4 x5 x6) (x8: @T8 x0 x1 x2 x3 x4 x5 x6 x7) (x9: @T9 x0 x1 x2 x3 x4 x5 x6 x7 x8) (x10: @T10 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9), Type.
Local Notation rel := (rel12 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11).
Variable gf: rel -> rel.
Hypothesis gf_mon: monotone12 gf.
Inductive sound12 (clo: rel -> rel): Prop :=
| sound12_intro
(MON: monotone12 clo)
(SOUND:
forall r (PFIX: r <12= gf (clo r)),
r <12= paco12 gf bot12)
.
Hint Constructors sound12.
Structure respectful12 (clo: rel -> rel) : Prop :=
respectful12_intro {
MON: monotone12 clo;
RESPECTFUL:
forall l r (LE: l <12= r) (GF: l <12= gf r),
clo l <12= gf (clo r);
}.
Hint Constructors respectful12.
Inductive gres12 (r: rel) e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 : Prop :=
| gres12_intro
clo
(RES: respectful12 clo)
(CLO: clo r e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11)
.
Hint Constructors gres12.
Lemma gfclo12_mon: forall clo, sound12 clo -> monotone12 (compose gf clo).
Proof.
intros; destruct H; red; intros.
eapply gf_mon; [apply IN|intros; eapply MON0; [apply PR|apply LE]].
Qed.
Hint Resolve gfclo12_mon : paco.
Lemma sound12_is_gf: forall clo (UPTO: sound12 clo),
paco12 (compose gf clo) bot12 <12= paco12 gf bot12.
Proof.
intros. _punfold PR; [|apply gfclo12_mon, UPTO]. edestruct UPTO.
eapply (SOUND (paco12 (compose gf clo) bot12)).
- intros. _punfold PR0; [|apply gfclo12_mon, UPTO].
eapply (gfclo12_mon UPTO); [apply PR0| intros; destruct PR1; [apply H|destruct H]].
- pfold. apply PR.
Qed.
Lemma respectful12_is_sound12: respectful12 <1= sound12.
Proof.
intro clo.
set (rclo := fix rclo clo n (r: rel) :=
match n with
| 0 => r
| S n' => rclo clo n' r \12/ clo (rclo clo n' r)
end).
intros. destruct PR. econstructor; [apply MON0|].
intros. set (rr e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 := exists n, rclo clo n r e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11).
assert (rr x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11) by (exists 0; apply PR); clear PR.
cut (forall n, rclo clo n r <12= gf (rclo clo (S n) r)).
{ intro X; revert x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 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 respectful12_compose
clo0 clo1
(RES0: respectful12 clo0)
(RES1: respectful12 clo1):
respectful12 (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 grespectful12_mon: monotone12 gres12.
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 grespectful12_respectful12: respectful12 gres12.
Proof.
econstructor; [apply grespectful12_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 gfgres12_mon: monotone12 (compose gf gres12).
Proof.
destruct grespectful12_respectful12.
unfold monotone12. intros. eapply gf_mon; [eapply IN|].
intros. eapply MON0;[apply PR|apply LE].
Qed.
Hint Resolve gfgres12_mon : paco.
Lemma grespectful12_greatest: forall clo (RES: respectful12 clo), clo <13= gres12.
Proof. intros. econstructor;[apply RES|apply PR]. Qed.
Lemma grespectful12_incl: forall r, r <12= gres12 r.
Proof.
intros; eexists (fun x => x).
- econstructor.
+ red; intros; apply LE, IN.
+ intros; apply GF, PR0.
- apply PR.
Qed.
Hint Resolve grespectful12_incl.
Lemma grespectful12_compose: forall clo (RES: respectful12 clo) r,
clo (gres12 r) <12= gres12 r.
Proof.
intros; eapply grespectful12_greatest with (clo := compose clo gres12); [|apply PR].
apply respectful12_compose; [apply RES|apply grespectful12_respectful12].
Qed.
Lemma grespectful12_incl_rev: forall r,
gres12 (paco12 (compose gf gres12) r) <12= paco12 (compose gf gres12) r.
Proof.
intro r; pcofix CIH; intros; pfold.
eapply gf_mon, grespectful12_compose, grespectful12_respectful12.
destruct grespectful12_respectful12; eapply RESPECTFUL0, PR; intros; [apply grespectful12_incl; right; apply CIH, grespectful12_incl, PR0|].
_punfold PR0; [|apply gfgres12_mon].
eapply gfgres12_mon; [apply PR0|].
intros; destruct PR1.
- left. eapply paco12_mon; [apply H| apply CIH0].
- right. eapply CIH0, H.
Qed.
Inductive rclo12 clo (r: rel): rel :=
| rclo12_incl
e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11
(R: r e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11):
@rclo12 clo r e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11
| rclo12_step'
r' e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11
(R': r' <12= rclo12 clo r)
(CLOR':clo r' e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11):
@rclo12 clo r e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11
| rclo12_gf
r' e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11
(R': r' <12= rclo12 clo r)
(CLOR':@gf r' e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11):
@rclo12 clo r e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11
.
Lemma rclo12_mon clo:
monotone12 (rclo12 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 rclo12_mon: paco.
Lemma rclo12_base
clo
(MON: monotone12 clo):
clo <13= rclo12 clo.
Proof.
intros. econstructor 2; [intros; apply PR0|].
eapply MON; [apply PR|intros; constructor; apply PR0].
Qed.
Lemma rclo12_step
(clo: rel -> rel) r:
clo (rclo12 clo r) <12= rclo12 clo r.
Proof.
intros. econstructor 2; [intros; apply PR0|apply PR].
Qed.
Lemma rclo12_rclo12
clo r
(MON: monotone12 clo):
rclo12 clo (rclo12 clo r) <12= rclo12 clo r.
Proof.
intros. induction PR.
- eapply R.
- econstructor 2; [eapply H | eapply CLOR'].
- econstructor 3; [eapply H | eapply CLOR'].
Qed.
Structure weak_respectful12 (clo: rel -> rel) : Prop :=
weak_respectful12_intro {
WEAK_MON: monotone12 clo;
WEAK_RESPECTFUL:
forall l r (LE: l <12= r) (GF: l <12= gf r),
clo l <12= gf (rclo12 clo r);
}.
Hint Constructors weak_respectful12.
Lemma weak_respectful12_respectful12
clo (RES: weak_respectful12 clo):
respectful12 (rclo12 clo).
Proof.
inversion RES. econstructor; [eapply rclo12_mon|]. intros.
induction PR; intros.
- eapply gf_mon; [apply GF, R|]. intros.
apply rclo12_incl. apply PR.
- eapply gf_mon.
+ eapply WEAK_RESPECTFUL0; [|apply H|apply CLOR'].
intros. eapply rclo12_mon; [apply R', PR|apply LE].
+ intros. apply rclo12_rclo12;[apply WEAK_MON0|apply PR].
- eapply gf_mon; [apply CLOR'|].
intros. eapply rclo12_mon; [apply R', PR| apply LE].
Qed.
Lemma upto12_init:
paco12 (compose gf gres12) bot12 <12= paco12 gf bot12.
Proof.
apply sound12_is_gf.
apply respectful12_is_sound12.
apply grespectful12_respectful12.
Qed.
Lemma upto12_final:
paco12 gf <13= paco12 (compose gf gres12).
Proof.
pcofix CIH. intros. _punfold PR; [|apply gf_mon]. pfold.
eapply gf_mon; [|apply grespectful12_incl].
eapply gf_mon; [apply PR|]. intros. right.
inversion PR0; [apply CIH, H | apply CIH0, H].
Qed.
Lemma upto12_step
r clo (RES: weak_respectful12 clo):
clo (paco12 (compose gf gres12) r) <12= paco12 (compose gf gres12) r.
Proof.
intros. apply grespectful12_incl_rev.
assert (RES' := weak_respectful12_respectful12 RES).
eapply grespectful12_greatest; [apply RES'|].
eapply rclo12_base; [apply RES|].
inversion RES. apply PR.
Qed.
Lemma upto12_step_under
r clo (RES: weak_respectful12 clo):
clo (gres12 r) <12= gres12 r.
Proof.
intros. apply weak_respectful12_respectful12 in RES.
eapply grespectful12_compose; [apply RES|].
econstructor 2; [intros; constructor 1; apply PR0 | apply PR].
Qed.
End Respectful12.
Lemma grespectful12_impl T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 (gf gf': rel12 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 -> rel12 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11) r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11
(PR: gres12 gf r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11)
(EQ: forall r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11, gf r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 <-> gf' r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11):
gres12 gf' r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11.
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 grespectful12_iff T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 (gf gf': rel12 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 -> rel12 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11) r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11
(EQ: forall r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11, gf r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 <-> gf' r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11):
gres12 gf r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 <-> gres12 gf' r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11.
Proof.
split; intros.
- eapply grespectful12_impl; [apply H | apply EQ].
- eapply grespectful12_impl; [apply H | split; apply EQ].
Qed.
Hint Constructors sound12.
Hint Constructors respectful12.
Hint Constructors gres12.
Hint Resolve gfclo12_mon : paco.
Hint Resolve gfgres12_mon : paco.
Hint Resolve grespectful12_incl.
Hint Resolve rclo12_mon: paco.
Hint Constructors weak_respectful12.
Ltac pupto12_init := eapply upto12_init; [eauto with paco|].
Ltac pupto12_final := first [eapply upto12_final; [eauto with paco|] | eapply grespectful12_incl].
Ltac pupto12 H := first [eapply upto12_step|eapply upto12_step_under]; [|eapply H|]; [eauto with paco|].