Library Paco.paco3_upto
Require Export Program.Basics. Open Scope program_scope.
Require Import paco3.
Set Implicit Arguments.
Section Respectful3.
Variable T0 : Type.
Variable T1 : forall (x0: @T0), Type.
Variable T2 : forall (x0: @T0) (x1: @T1 x0), Type.
Local Notation rel := (rel3 T0 T1 T2).
Variable gf: rel -> rel.
Hypothesis gf_mon: monotone3 gf.
Inductive sound3 (clo: rel -> rel): Prop :=
| sound3_intro
(MON: monotone3 clo)
(SOUND:
forall r (PFIX: r <3= gf (clo r)),
r <3= paco3 gf bot3)
.
Hint Constructors sound3.
Structure respectful3 (clo: rel -> rel) : Prop :=
respectful3_intro {
MON: monotone3 clo;
RESPECTFUL:
forall l r (LE: l <3= r) (GF: l <3= gf r),
clo l <3= gf (clo r);
}.
Hint Constructors respectful3.
Inductive gres3 (r: rel) e0 e1 e2 : Prop :=
| gres3_intro
clo
(RES: respectful3 clo)
(CLO: clo r e0 e1 e2)
.
Hint Constructors gres3.
Lemma gfclo3_mon: forall clo, sound3 clo -> monotone3 (compose gf clo).
Proof.
intros; destruct H; red; intros.
eapply gf_mon; [apply IN|intros; eapply MON0; [apply PR|apply LE]].
Qed.
Hint Resolve gfclo3_mon : paco.
Lemma sound3_is_gf: forall clo (UPTO: sound3 clo),
paco3 (compose gf clo) bot3 <3= paco3 gf bot3.
Proof.
intros. _punfold PR; [|apply gfclo3_mon, UPTO]. edestruct UPTO.
eapply (SOUND (paco3 (compose gf clo) bot3)).
- intros. _punfold PR0; [|apply gfclo3_mon, UPTO].
eapply (gfclo3_mon UPTO); [apply PR0| intros; destruct PR1; [apply H|destruct H]].
- pfold. apply PR.
Qed.
Lemma respectful3_is_sound3: respectful3 <1= sound3.
Proof.
intro clo.
set (rclo := fix rclo clo n (r: rel) :=
match n with
| 0 => r
| S n' => rclo clo n' r \3/ clo (rclo clo n' r)
end).
intros. destruct PR. econstructor; [apply MON0|].
intros. set (rr e0 e1 e2 := exists n, rclo clo n r e0 e1 e2).
assert (rr x0 x1 x2) by (exists 0; apply PR); clear PR.
cut (forall n, rclo clo n r <3= gf (rclo clo (S n) r)).
{ intro X; revert x0 x1 x2 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 respectful3_compose
clo0 clo1
(RES0: respectful3 clo0)
(RES1: respectful3 clo1):
respectful3 (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 grespectful3_mon: monotone3 gres3.
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 grespectful3_respectful3: respectful3 gres3.
Proof.
econstructor; [apply grespectful3_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 gfgres3_mon: monotone3 (compose gf gres3).
Proof.
destruct grespectful3_respectful3.
unfold monotone3. intros. eapply gf_mon; [eapply IN|].
intros. eapply MON0;[apply PR|apply LE].
Qed.
Hint Resolve gfgres3_mon : paco.
Lemma grespectful3_greatest: forall clo (RES: respectful3 clo), clo <4= gres3.
Proof. intros. econstructor;[apply RES|apply PR]. Qed.
Lemma grespectful3_incl: forall r, r <3= gres3 r.
Proof.
intros; eexists (fun x => x).
- econstructor.
+ red; intros; apply LE, IN.
+ intros; apply GF, PR0.
- apply PR.
Qed.
Hint Resolve grespectful3_incl.
Lemma grespectful3_compose: forall clo (RES: respectful3 clo) r,
clo (gres3 r) <3= gres3 r.
Proof.
intros; eapply grespectful3_greatest with (clo := compose clo gres3); [|apply PR].
apply respectful3_compose; [apply RES|apply grespectful3_respectful3].
Qed.
Lemma grespectful3_incl_rev: forall r,
gres3 (paco3 (compose gf gres3) r) <3= paco3 (compose gf gres3) r.
Proof.
intro r; pcofix CIH; intros; pfold.
eapply gf_mon, grespectful3_compose, grespectful3_respectful3.
destruct grespectful3_respectful3; eapply RESPECTFUL0, PR; intros; [apply grespectful3_incl; right; apply CIH, grespectful3_incl, PR0|].
_punfold PR0; [|apply gfgres3_mon].
eapply gfgres3_mon; [apply PR0|].
intros; destruct PR1.
- left. eapply paco3_mon; [apply H| apply CIH0].
- right. eapply CIH0, H.
Qed.
Inductive rclo3 clo (r: rel): rel :=
| rclo3_incl
e0 e1 e2
(R: r e0 e1 e2):
@rclo3 clo r e0 e1 e2
| rclo3_step'
r' e0 e1 e2
(R': r' <3= rclo3 clo r)
(CLOR':clo r' e0 e1 e2):
@rclo3 clo r e0 e1 e2
| rclo3_gf
r' e0 e1 e2
(R': r' <3= rclo3 clo r)
(CLOR':@gf r' e0 e1 e2):
@rclo3 clo r e0 e1 e2
.
Lemma rclo3_mon clo:
monotone3 (rclo3 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 rclo3_mon: paco.
Lemma rclo3_base
clo
(MON: monotone3 clo):
clo <4= rclo3 clo.
Proof.
intros. econstructor 2; [intros; apply PR0|].
eapply MON; [apply PR|intros; constructor; apply PR0].
Qed.
Lemma rclo3_step
(clo: rel -> rel) r:
clo (rclo3 clo r) <3= rclo3 clo r.
Proof.
intros. econstructor 2; [intros; apply PR0|apply PR].
Qed.
Lemma rclo3_rclo3
clo r
(MON: monotone3 clo):
rclo3 clo (rclo3 clo r) <3= rclo3 clo r.
Proof.
intros. induction PR.
- eapply R.
- econstructor 2; [eapply H | eapply CLOR'].
- econstructor 3; [eapply H | eapply CLOR'].
Qed.
Structure weak_respectful3 (clo: rel -> rel) : Prop :=
weak_respectful3_intro {
WEAK_MON: monotone3 clo;
WEAK_RESPECTFUL:
forall l r (LE: l <3= r) (GF: l <3= gf r),
clo l <3= gf (rclo3 clo r);
}.
Hint Constructors weak_respectful3.
Lemma weak_respectful3_respectful3
clo (RES: weak_respectful3 clo):
respectful3 (rclo3 clo).
Proof.
inversion RES. econstructor; [eapply rclo3_mon|]. intros.
induction PR; intros.
- eapply gf_mon; [apply GF, R|]. intros.
apply rclo3_incl. apply PR.
- eapply gf_mon.
+ eapply WEAK_RESPECTFUL0; [|apply H|apply CLOR'].
intros. eapply rclo3_mon; [apply R', PR|apply LE].
+ intros. apply rclo3_rclo3;[apply WEAK_MON0|apply PR].
- eapply gf_mon; [apply CLOR'|].
intros. eapply rclo3_mon; [apply R', PR| apply LE].
Qed.
Lemma upto3_init:
paco3 (compose gf gres3) bot3 <3= paco3 gf bot3.
Proof.
apply sound3_is_gf.
apply respectful3_is_sound3.
apply grespectful3_respectful3.
Qed.
Lemma upto3_final:
paco3 gf <4= paco3 (compose gf gres3).
Proof.
pcofix CIH. intros. _punfold PR; [|apply gf_mon]. pfold.
eapply gf_mon; [|apply grespectful3_incl].
eapply gf_mon; [apply PR|]. intros. right.
inversion PR0; [apply CIH, H | apply CIH0, H].
Qed.
Lemma upto3_step
r clo (RES: weak_respectful3 clo):
clo (paco3 (compose gf gres3) r) <3= paco3 (compose gf gres3) r.
Proof.
intros. apply grespectful3_incl_rev.
assert (RES' := weak_respectful3_respectful3 RES).
eapply grespectful3_greatest; [apply RES'|].
eapply rclo3_base; [apply RES|].
inversion RES. apply PR.
Qed.
Lemma upto3_step_under
r clo (RES: weak_respectful3 clo):
clo (gres3 r) <3= gres3 r.
Proof.
intros. apply weak_respectful3_respectful3 in RES.
eapply grespectful3_compose; [apply RES|].
econstructor 2; [intros; constructor 1; apply PR0 | apply PR].
Qed.
End Respectful3.
Lemma grespectful3_impl T0 T1 T2 (gf gf': rel3 T0 T1 T2 -> rel3 T0 T1 T2) r x0 x1 x2
(PR: gres3 gf r x0 x1 x2)
(EQ: forall r x0 x1 x2, gf r x0 x1 x2 <-> gf' r x0 x1 x2):
gres3 gf' r x0 x1 x2.
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 grespectful3_iff T0 T1 T2 (gf gf': rel3 T0 T1 T2 -> rel3 T0 T1 T2) r x0 x1 x2
(EQ: forall r x0 x1 x2, gf r x0 x1 x2 <-> gf' r x0 x1 x2):
gres3 gf r x0 x1 x2 <-> gres3 gf' r x0 x1 x2.
Proof.
split; intros.
- eapply grespectful3_impl; [apply H | apply EQ].
- eapply grespectful3_impl; [apply H | split; apply EQ].
Qed.
Hint Constructors sound3.
Hint Constructors respectful3.
Hint Constructors gres3.
Hint Resolve gfclo3_mon : paco.
Hint Resolve gfgres3_mon : paco.
Hint Resolve grespectful3_incl.
Hint Resolve rclo3_mon: paco.
Hint Constructors weak_respectful3.
Ltac pupto3_init := eapply upto3_init; [eauto with paco|].
Ltac pupto3_final := first [eapply upto3_final; [eauto with paco|] | eapply grespectful3_incl].
Ltac pupto3 H := first [eapply upto3_step|eapply upto3_step_under]; [|eapply H|]; [eauto with paco|].
Require Import paco3.
Set Implicit Arguments.
Section Respectful3.
Variable T0 : Type.
Variable T1 : forall (x0: @T0), Type.
Variable T2 : forall (x0: @T0) (x1: @T1 x0), Type.
Local Notation rel := (rel3 T0 T1 T2).
Variable gf: rel -> rel.
Hypothesis gf_mon: monotone3 gf.
Inductive sound3 (clo: rel -> rel): Prop :=
| sound3_intro
(MON: monotone3 clo)
(SOUND:
forall r (PFIX: r <3= gf (clo r)),
r <3= paco3 gf bot3)
.
Hint Constructors sound3.
Structure respectful3 (clo: rel -> rel) : Prop :=
respectful3_intro {
MON: monotone3 clo;
RESPECTFUL:
forall l r (LE: l <3= r) (GF: l <3= gf r),
clo l <3= gf (clo r);
}.
Hint Constructors respectful3.
Inductive gres3 (r: rel) e0 e1 e2 : Prop :=
| gres3_intro
clo
(RES: respectful3 clo)
(CLO: clo r e0 e1 e2)
.
Hint Constructors gres3.
Lemma gfclo3_mon: forall clo, sound3 clo -> monotone3 (compose gf clo).
Proof.
intros; destruct H; red; intros.
eapply gf_mon; [apply IN|intros; eapply MON0; [apply PR|apply LE]].
Qed.
Hint Resolve gfclo3_mon : paco.
Lemma sound3_is_gf: forall clo (UPTO: sound3 clo),
paco3 (compose gf clo) bot3 <3= paco3 gf bot3.
Proof.
intros. _punfold PR; [|apply gfclo3_mon, UPTO]. edestruct UPTO.
eapply (SOUND (paco3 (compose gf clo) bot3)).
- intros. _punfold PR0; [|apply gfclo3_mon, UPTO].
eapply (gfclo3_mon UPTO); [apply PR0| intros; destruct PR1; [apply H|destruct H]].
- pfold. apply PR.
Qed.
Lemma respectful3_is_sound3: respectful3 <1= sound3.
Proof.
intro clo.
set (rclo := fix rclo clo n (r: rel) :=
match n with
| 0 => r
| S n' => rclo clo n' r \3/ clo (rclo clo n' r)
end).
intros. destruct PR. econstructor; [apply MON0|].
intros. set (rr e0 e1 e2 := exists n, rclo clo n r e0 e1 e2).
assert (rr x0 x1 x2) by (exists 0; apply PR); clear PR.
cut (forall n, rclo clo n r <3= gf (rclo clo (S n) r)).
{ intro X; revert x0 x1 x2 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 respectful3_compose
clo0 clo1
(RES0: respectful3 clo0)
(RES1: respectful3 clo1):
respectful3 (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 grespectful3_mon: monotone3 gres3.
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 grespectful3_respectful3: respectful3 gres3.
Proof.
econstructor; [apply grespectful3_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 gfgres3_mon: monotone3 (compose gf gres3).
Proof.
destruct grespectful3_respectful3.
unfold monotone3. intros. eapply gf_mon; [eapply IN|].
intros. eapply MON0;[apply PR|apply LE].
Qed.
Hint Resolve gfgres3_mon : paco.
Lemma grespectful3_greatest: forall clo (RES: respectful3 clo), clo <4= gres3.
Proof. intros. econstructor;[apply RES|apply PR]. Qed.
Lemma grespectful3_incl: forall r, r <3= gres3 r.
Proof.
intros; eexists (fun x => x).
- econstructor.
+ red; intros; apply LE, IN.
+ intros; apply GF, PR0.
- apply PR.
Qed.
Hint Resolve grespectful3_incl.
Lemma grespectful3_compose: forall clo (RES: respectful3 clo) r,
clo (gres3 r) <3= gres3 r.
Proof.
intros; eapply grespectful3_greatest with (clo := compose clo gres3); [|apply PR].
apply respectful3_compose; [apply RES|apply grespectful3_respectful3].
Qed.
Lemma grespectful3_incl_rev: forall r,
gres3 (paco3 (compose gf gres3) r) <3= paco3 (compose gf gres3) r.
Proof.
intro r; pcofix CIH; intros; pfold.
eapply gf_mon, grespectful3_compose, grespectful3_respectful3.
destruct grespectful3_respectful3; eapply RESPECTFUL0, PR; intros; [apply grespectful3_incl; right; apply CIH, grespectful3_incl, PR0|].
_punfold PR0; [|apply gfgres3_mon].
eapply gfgres3_mon; [apply PR0|].
intros; destruct PR1.
- left. eapply paco3_mon; [apply H| apply CIH0].
- right. eapply CIH0, H.
Qed.
Inductive rclo3 clo (r: rel): rel :=
| rclo3_incl
e0 e1 e2
(R: r e0 e1 e2):
@rclo3 clo r e0 e1 e2
| rclo3_step'
r' e0 e1 e2
(R': r' <3= rclo3 clo r)
(CLOR':clo r' e0 e1 e2):
@rclo3 clo r e0 e1 e2
| rclo3_gf
r' e0 e1 e2
(R': r' <3= rclo3 clo r)
(CLOR':@gf r' e0 e1 e2):
@rclo3 clo r e0 e1 e2
.
Lemma rclo3_mon clo:
monotone3 (rclo3 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 rclo3_mon: paco.
Lemma rclo3_base
clo
(MON: monotone3 clo):
clo <4= rclo3 clo.
Proof.
intros. econstructor 2; [intros; apply PR0|].
eapply MON; [apply PR|intros; constructor; apply PR0].
Qed.
Lemma rclo3_step
(clo: rel -> rel) r:
clo (rclo3 clo r) <3= rclo3 clo r.
Proof.
intros. econstructor 2; [intros; apply PR0|apply PR].
Qed.
Lemma rclo3_rclo3
clo r
(MON: monotone3 clo):
rclo3 clo (rclo3 clo r) <3= rclo3 clo r.
Proof.
intros. induction PR.
- eapply R.
- econstructor 2; [eapply H | eapply CLOR'].
- econstructor 3; [eapply H | eapply CLOR'].
Qed.
Structure weak_respectful3 (clo: rel -> rel) : Prop :=
weak_respectful3_intro {
WEAK_MON: monotone3 clo;
WEAK_RESPECTFUL:
forall l r (LE: l <3= r) (GF: l <3= gf r),
clo l <3= gf (rclo3 clo r);
}.
Hint Constructors weak_respectful3.
Lemma weak_respectful3_respectful3
clo (RES: weak_respectful3 clo):
respectful3 (rclo3 clo).
Proof.
inversion RES. econstructor; [eapply rclo3_mon|]. intros.
induction PR; intros.
- eapply gf_mon; [apply GF, R|]. intros.
apply rclo3_incl. apply PR.
- eapply gf_mon.
+ eapply WEAK_RESPECTFUL0; [|apply H|apply CLOR'].
intros. eapply rclo3_mon; [apply R', PR|apply LE].
+ intros. apply rclo3_rclo3;[apply WEAK_MON0|apply PR].
- eapply gf_mon; [apply CLOR'|].
intros. eapply rclo3_mon; [apply R', PR| apply LE].
Qed.
Lemma upto3_init:
paco3 (compose gf gres3) bot3 <3= paco3 gf bot3.
Proof.
apply sound3_is_gf.
apply respectful3_is_sound3.
apply grespectful3_respectful3.
Qed.
Lemma upto3_final:
paco3 gf <4= paco3 (compose gf gres3).
Proof.
pcofix CIH. intros. _punfold PR; [|apply gf_mon]. pfold.
eapply gf_mon; [|apply grespectful3_incl].
eapply gf_mon; [apply PR|]. intros. right.
inversion PR0; [apply CIH, H | apply CIH0, H].
Qed.
Lemma upto3_step
r clo (RES: weak_respectful3 clo):
clo (paco3 (compose gf gres3) r) <3= paco3 (compose gf gres3) r.
Proof.
intros. apply grespectful3_incl_rev.
assert (RES' := weak_respectful3_respectful3 RES).
eapply grespectful3_greatest; [apply RES'|].
eapply rclo3_base; [apply RES|].
inversion RES. apply PR.
Qed.
Lemma upto3_step_under
r clo (RES: weak_respectful3 clo):
clo (gres3 r) <3= gres3 r.
Proof.
intros. apply weak_respectful3_respectful3 in RES.
eapply grespectful3_compose; [apply RES|].
econstructor 2; [intros; constructor 1; apply PR0 | apply PR].
Qed.
End Respectful3.
Lemma grespectful3_impl T0 T1 T2 (gf gf': rel3 T0 T1 T2 -> rel3 T0 T1 T2) r x0 x1 x2
(PR: gres3 gf r x0 x1 x2)
(EQ: forall r x0 x1 x2, gf r x0 x1 x2 <-> gf' r x0 x1 x2):
gres3 gf' r x0 x1 x2.
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 grespectful3_iff T0 T1 T2 (gf gf': rel3 T0 T1 T2 -> rel3 T0 T1 T2) r x0 x1 x2
(EQ: forall r x0 x1 x2, gf r x0 x1 x2 <-> gf' r x0 x1 x2):
gres3 gf r x0 x1 x2 <-> gres3 gf' r x0 x1 x2.
Proof.
split; intros.
- eapply grespectful3_impl; [apply H | apply EQ].
- eapply grespectful3_impl; [apply H | split; apply EQ].
Qed.
Hint Constructors sound3.
Hint Constructors respectful3.
Hint Constructors gres3.
Hint Resolve gfclo3_mon : paco.
Hint Resolve gfgres3_mon : paco.
Hint Resolve grespectful3_incl.
Hint Resolve rclo3_mon: paco.
Hint Constructors weak_respectful3.
Ltac pupto3_init := eapply upto3_init; [eauto with paco|].
Ltac pupto3_final := first [eapply upto3_final; [eauto with paco|] | eapply grespectful3_incl].
Ltac pupto3 H := first [eapply upto3_step|eapply upto3_step_under]; [|eapply H|]; [eauto with paco|].