Library Paco.paco14_upto
Require Export Program.Basics. Open Scope program_scope.
Require Import paco14.
Set Implicit Arguments.
Section Respectful14.
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.
Variable T12 : 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) (x11: @T11 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10), Type.
Variable T13 : 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) (x11: @T11 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10) (x12: @T12 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11), Type.
Local Notation rel := (rel14 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 T13).
Variable gf: rel -> rel.
Hypothesis gf_mon: monotone14 gf.
Inductive sound14 (clo: rel -> rel): Prop :=
| sound14_intro
(MON: monotone14 clo)
(SOUND:
forall r (PFIX: r <14= gf (clo r)),
r <14= paco14 gf bot14)
.
Hint Constructors sound14.
Structure respectful14 (clo: rel -> rel) : Prop :=
respectful14_intro {
MON: monotone14 clo;
RESPECTFUL:
forall l r (LE: l <14= r) (GF: l <14= gf r),
clo l <14= gf (clo r);
}.
Hint Constructors respectful14.
Inductive gres14 (r: rel) e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13 : Prop :=
| gres14_intro
clo
(RES: respectful14 clo)
(CLO: clo r e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13)
.
Hint Constructors gres14.
Lemma gfclo14_mon: forall clo, sound14 clo -> monotone14 (compose gf clo).
Proof.
intros; destruct H; red; intros.
eapply gf_mon; [apply IN|intros; eapply MON0; [apply PR|apply LE]].
Qed.
Hint Resolve gfclo14_mon : paco.
Lemma sound14_is_gf: forall clo (UPTO: sound14 clo),
paco14 (compose gf clo) bot14 <14= paco14 gf bot14.
Proof.
intros. _punfold PR; [|apply gfclo14_mon, UPTO]. edestruct UPTO.
eapply (SOUND (paco14 (compose gf clo) bot14)).
- intros. _punfold PR0; [|apply gfclo14_mon, UPTO].
eapply (gfclo14_mon UPTO); [apply PR0| intros; destruct PR1; [apply H|destruct H]].
- pfold. apply PR.
Qed.
Lemma respectful14_is_sound14: respectful14 <1= sound14.
Proof.
intro clo.
set (rclo := fix rclo clo n (r: rel) :=
match n with
| 0 => r
| S n' => rclo clo n' r \14/ 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 e12 e13 := exists n, rclo clo n r e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13).
assert (rr x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13) by (exists 0; apply PR); clear PR.
cut (forall n, rclo clo n r <14= gf (rclo clo (S n) r)).
{ intro X; revert x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 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 respectful14_compose
clo0 clo1
(RES0: respectful14 clo0)
(RES1: respectful14 clo1):
respectful14 (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 grespectful14_mon: monotone14 gres14.
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 grespectful14_respectful14: respectful14 gres14.
Proof.
econstructor; [apply grespectful14_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 gfgres14_mon: monotone14 (compose gf gres14).
Proof.
destruct grespectful14_respectful14.
unfold monotone14. intros. eapply gf_mon; [eapply IN|].
intros. eapply MON0;[apply PR|apply LE].
Qed.
Hint Resolve gfgres14_mon : paco.
Lemma grespectful14_greatest: forall clo (RES: respectful14 clo), clo <15= gres14.
Proof. intros. econstructor;[apply RES|apply PR]. Qed.
Lemma grespectful14_incl: forall r, r <14= gres14 r.
Proof.
intros; eexists (fun x => x).
- econstructor.
+ red; intros; apply LE, IN.
+ intros; apply GF, PR0.
- apply PR.
Qed.
Hint Resolve grespectful14_incl.
Lemma grespectful14_compose: forall clo (RES: respectful14 clo) r,
clo (gres14 r) <14= gres14 r.
Proof.
intros; eapply grespectful14_greatest with (clo := compose clo gres14); [|apply PR].
apply respectful14_compose; [apply RES|apply grespectful14_respectful14].
Qed.
Lemma grespectful14_incl_rev: forall r,
gres14 (paco14 (compose gf gres14) r) <14= paco14 (compose gf gres14) r.
Proof.
intro r; pcofix CIH; intros; pfold.
eapply gf_mon, grespectful14_compose, grespectful14_respectful14.
destruct grespectful14_respectful14; eapply RESPECTFUL0, PR; intros; [apply grespectful14_incl; right; apply CIH, grespectful14_incl, PR0|].
_punfold PR0; [|apply gfgres14_mon].
eapply gfgres14_mon; [apply PR0|].
intros; destruct PR1.
- left. eapply paco14_mon; [apply H| apply CIH0].
- right. eapply CIH0, H.
Qed.
Inductive rclo14 clo (r: rel): rel :=
| rclo14_incl
e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13
(R: r e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13):
@rclo14 clo r e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13
| rclo14_step'
r' e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13
(R': r' <14= rclo14 clo r)
(CLOR':clo r' e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13):
@rclo14 clo r e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13
| rclo14_gf
r' e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13
(R': r' <14= rclo14 clo r)
(CLOR':@gf r' e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13):
@rclo14 clo r e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13
.
Lemma rclo14_mon clo:
monotone14 (rclo14 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 rclo14_mon: paco.
Lemma rclo14_base
clo
(MON: monotone14 clo):
clo <15= rclo14 clo.
Proof.
intros. econstructor 2; [intros; apply PR0|].
eapply MON; [apply PR|intros; constructor; apply PR0].
Qed.
Lemma rclo14_step
(clo: rel -> rel) r:
clo (rclo14 clo r) <14= rclo14 clo r.
Proof.
intros. econstructor 2; [intros; apply PR0|apply PR].
Qed.
Lemma rclo14_rclo14
clo r
(MON: monotone14 clo):
rclo14 clo (rclo14 clo r) <14= rclo14 clo r.
Proof.
intros. induction PR.
- eapply R.
- econstructor 2; [eapply H | eapply CLOR'].
- econstructor 3; [eapply H | eapply CLOR'].
Qed.
Structure weak_respectful14 (clo: rel -> rel) : Prop :=
weak_respectful14_intro {
WEAK_MON: monotone14 clo;
WEAK_RESPECTFUL:
forall l r (LE: l <14= r) (GF: l <14= gf r),
clo l <14= gf (rclo14 clo r);
}.
Hint Constructors weak_respectful14.
Lemma weak_respectful14_respectful14
clo (RES: weak_respectful14 clo):
respectful14 (rclo14 clo).
Proof.
inversion RES. econstructor; [eapply rclo14_mon|]. intros.
induction PR; intros.
- eapply gf_mon; [apply GF, R|]. intros.
apply rclo14_incl. apply PR.
- eapply gf_mon.
+ eapply WEAK_RESPECTFUL0; [|apply H|apply CLOR'].
intros. eapply rclo14_mon; [apply R', PR|apply LE].
+ intros. apply rclo14_rclo14;[apply WEAK_MON0|apply PR].
- eapply gf_mon; [apply CLOR'|].
intros. eapply rclo14_mon; [apply R', PR| apply LE].
Qed.
Lemma upto14_init:
paco14 (compose gf gres14) bot14 <14= paco14 gf bot14.
Proof.
apply sound14_is_gf.
apply respectful14_is_sound14.
apply grespectful14_respectful14.
Qed.
Lemma upto14_final:
paco14 gf <15= paco14 (compose gf gres14).
Proof.
pcofix CIH. intros. _punfold PR; [|apply gf_mon]. pfold.
eapply gf_mon; [|apply grespectful14_incl].
eapply gf_mon; [apply PR|]. intros. right.
inversion PR0; [apply CIH, H | apply CIH0, H].
Qed.
Lemma upto14_step
r clo (RES: weak_respectful14 clo):
clo (paco14 (compose gf gres14) r) <14= paco14 (compose gf gres14) r.
Proof.
intros. apply grespectful14_incl_rev.
assert (RES' := weak_respectful14_respectful14 RES).
eapply grespectful14_greatest; [apply RES'|].
eapply rclo14_base; [apply RES|].
inversion RES. apply PR.
Qed.
Lemma upto14_step_under
r clo (RES: weak_respectful14 clo):
clo (gres14 r) <14= gres14 r.
Proof.
intros. apply weak_respectful14_respectful14 in RES.
eapply grespectful14_compose; [apply RES|].
econstructor 2; [intros; constructor 1; apply PR0 | apply PR].
Qed.
End Respectful14.
Lemma grespectful14_impl T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 T13 (gf gf': rel14 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 T13 -> rel14 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 T13) r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13
(PR: gres14 gf r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13)
(EQ: forall r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13, gf r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 <-> gf' r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13):
gres14 gf' r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13.
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 grespectful14_iff T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 T13 (gf gf': rel14 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 T13 -> rel14 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 T13) r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13
(EQ: forall r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13, gf r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 <-> gf' r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13):
gres14 gf r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 <-> gres14 gf' r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13.
Proof.
split; intros.
- eapply grespectful14_impl; [apply H | apply EQ].
- eapply grespectful14_impl; [apply H | split; apply EQ].
Qed.
Hint Constructors sound14.
Hint Constructors respectful14.
Hint Constructors gres14.
Hint Resolve gfclo14_mon : paco.
Hint Resolve gfgres14_mon : paco.
Hint Resolve grespectful14_incl.
Hint Resolve rclo14_mon: paco.
Hint Constructors weak_respectful14.
Ltac pupto14_init := eapply upto14_init; [eauto with paco|].
Ltac pupto14_final := first [eapply upto14_final; [eauto with paco|] | eapply grespectful14_incl].
Ltac pupto14 H := first [eapply upto14_step|eapply upto14_step_under]; [|eapply H|]; [eauto with paco|].
Require Import paco14.
Set Implicit Arguments.
Section Respectful14.
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.
Variable T12 : 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) (x11: @T11 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10), Type.
Variable T13 : 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) (x11: @T11 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10) (x12: @T12 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11), Type.
Local Notation rel := (rel14 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 T13).
Variable gf: rel -> rel.
Hypothesis gf_mon: monotone14 gf.
Inductive sound14 (clo: rel -> rel): Prop :=
| sound14_intro
(MON: monotone14 clo)
(SOUND:
forall r (PFIX: r <14= gf (clo r)),
r <14= paco14 gf bot14)
.
Hint Constructors sound14.
Structure respectful14 (clo: rel -> rel) : Prop :=
respectful14_intro {
MON: monotone14 clo;
RESPECTFUL:
forall l r (LE: l <14= r) (GF: l <14= gf r),
clo l <14= gf (clo r);
}.
Hint Constructors respectful14.
Inductive gres14 (r: rel) e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13 : Prop :=
| gres14_intro
clo
(RES: respectful14 clo)
(CLO: clo r e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13)
.
Hint Constructors gres14.
Lemma gfclo14_mon: forall clo, sound14 clo -> monotone14 (compose gf clo).
Proof.
intros; destruct H; red; intros.
eapply gf_mon; [apply IN|intros; eapply MON0; [apply PR|apply LE]].
Qed.
Hint Resolve gfclo14_mon : paco.
Lemma sound14_is_gf: forall clo (UPTO: sound14 clo),
paco14 (compose gf clo) bot14 <14= paco14 gf bot14.
Proof.
intros. _punfold PR; [|apply gfclo14_mon, UPTO]. edestruct UPTO.
eapply (SOUND (paco14 (compose gf clo) bot14)).
- intros. _punfold PR0; [|apply gfclo14_mon, UPTO].
eapply (gfclo14_mon UPTO); [apply PR0| intros; destruct PR1; [apply H|destruct H]].
- pfold. apply PR.
Qed.
Lemma respectful14_is_sound14: respectful14 <1= sound14.
Proof.
intro clo.
set (rclo := fix rclo clo n (r: rel) :=
match n with
| 0 => r
| S n' => rclo clo n' r \14/ 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 e12 e13 := exists n, rclo clo n r e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13).
assert (rr x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13) by (exists 0; apply PR); clear PR.
cut (forall n, rclo clo n r <14= gf (rclo clo (S n) r)).
{ intro X; revert x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 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 respectful14_compose
clo0 clo1
(RES0: respectful14 clo0)
(RES1: respectful14 clo1):
respectful14 (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 grespectful14_mon: monotone14 gres14.
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 grespectful14_respectful14: respectful14 gres14.
Proof.
econstructor; [apply grespectful14_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 gfgres14_mon: monotone14 (compose gf gres14).
Proof.
destruct grespectful14_respectful14.
unfold monotone14. intros. eapply gf_mon; [eapply IN|].
intros. eapply MON0;[apply PR|apply LE].
Qed.
Hint Resolve gfgres14_mon : paco.
Lemma grespectful14_greatest: forall clo (RES: respectful14 clo), clo <15= gres14.
Proof. intros. econstructor;[apply RES|apply PR]. Qed.
Lemma grespectful14_incl: forall r, r <14= gres14 r.
Proof.
intros; eexists (fun x => x).
- econstructor.
+ red; intros; apply LE, IN.
+ intros; apply GF, PR0.
- apply PR.
Qed.
Hint Resolve grespectful14_incl.
Lemma grespectful14_compose: forall clo (RES: respectful14 clo) r,
clo (gres14 r) <14= gres14 r.
Proof.
intros; eapply grespectful14_greatest with (clo := compose clo gres14); [|apply PR].
apply respectful14_compose; [apply RES|apply grespectful14_respectful14].
Qed.
Lemma grespectful14_incl_rev: forall r,
gres14 (paco14 (compose gf gres14) r) <14= paco14 (compose gf gres14) r.
Proof.
intro r; pcofix CIH; intros; pfold.
eapply gf_mon, grespectful14_compose, grespectful14_respectful14.
destruct grespectful14_respectful14; eapply RESPECTFUL0, PR; intros; [apply grespectful14_incl; right; apply CIH, grespectful14_incl, PR0|].
_punfold PR0; [|apply gfgres14_mon].
eapply gfgres14_mon; [apply PR0|].
intros; destruct PR1.
- left. eapply paco14_mon; [apply H| apply CIH0].
- right. eapply CIH0, H.
Qed.
Inductive rclo14 clo (r: rel): rel :=
| rclo14_incl
e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13
(R: r e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13):
@rclo14 clo r e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13
| rclo14_step'
r' e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13
(R': r' <14= rclo14 clo r)
(CLOR':clo r' e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13):
@rclo14 clo r e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13
| rclo14_gf
r' e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13
(R': r' <14= rclo14 clo r)
(CLOR':@gf r' e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13):
@rclo14 clo r e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13
.
Lemma rclo14_mon clo:
monotone14 (rclo14 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 rclo14_mon: paco.
Lemma rclo14_base
clo
(MON: monotone14 clo):
clo <15= rclo14 clo.
Proof.
intros. econstructor 2; [intros; apply PR0|].
eapply MON; [apply PR|intros; constructor; apply PR0].
Qed.
Lemma rclo14_step
(clo: rel -> rel) r:
clo (rclo14 clo r) <14= rclo14 clo r.
Proof.
intros. econstructor 2; [intros; apply PR0|apply PR].
Qed.
Lemma rclo14_rclo14
clo r
(MON: monotone14 clo):
rclo14 clo (rclo14 clo r) <14= rclo14 clo r.
Proof.
intros. induction PR.
- eapply R.
- econstructor 2; [eapply H | eapply CLOR'].
- econstructor 3; [eapply H | eapply CLOR'].
Qed.
Structure weak_respectful14 (clo: rel -> rel) : Prop :=
weak_respectful14_intro {
WEAK_MON: monotone14 clo;
WEAK_RESPECTFUL:
forall l r (LE: l <14= r) (GF: l <14= gf r),
clo l <14= gf (rclo14 clo r);
}.
Hint Constructors weak_respectful14.
Lemma weak_respectful14_respectful14
clo (RES: weak_respectful14 clo):
respectful14 (rclo14 clo).
Proof.
inversion RES. econstructor; [eapply rclo14_mon|]. intros.
induction PR; intros.
- eapply gf_mon; [apply GF, R|]. intros.
apply rclo14_incl. apply PR.
- eapply gf_mon.
+ eapply WEAK_RESPECTFUL0; [|apply H|apply CLOR'].
intros. eapply rclo14_mon; [apply R', PR|apply LE].
+ intros. apply rclo14_rclo14;[apply WEAK_MON0|apply PR].
- eapply gf_mon; [apply CLOR'|].
intros. eapply rclo14_mon; [apply R', PR| apply LE].
Qed.
Lemma upto14_init:
paco14 (compose gf gres14) bot14 <14= paco14 gf bot14.
Proof.
apply sound14_is_gf.
apply respectful14_is_sound14.
apply grespectful14_respectful14.
Qed.
Lemma upto14_final:
paco14 gf <15= paco14 (compose gf gres14).
Proof.
pcofix CIH. intros. _punfold PR; [|apply gf_mon]. pfold.
eapply gf_mon; [|apply grespectful14_incl].
eapply gf_mon; [apply PR|]. intros. right.
inversion PR0; [apply CIH, H | apply CIH0, H].
Qed.
Lemma upto14_step
r clo (RES: weak_respectful14 clo):
clo (paco14 (compose gf gres14) r) <14= paco14 (compose gf gres14) r.
Proof.
intros. apply grespectful14_incl_rev.
assert (RES' := weak_respectful14_respectful14 RES).
eapply grespectful14_greatest; [apply RES'|].
eapply rclo14_base; [apply RES|].
inversion RES. apply PR.
Qed.
Lemma upto14_step_under
r clo (RES: weak_respectful14 clo):
clo (gres14 r) <14= gres14 r.
Proof.
intros. apply weak_respectful14_respectful14 in RES.
eapply grespectful14_compose; [apply RES|].
econstructor 2; [intros; constructor 1; apply PR0 | apply PR].
Qed.
End Respectful14.
Lemma grespectful14_impl T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 T13 (gf gf': rel14 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 T13 -> rel14 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 T13) r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13
(PR: gres14 gf r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13)
(EQ: forall r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13, gf r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 <-> gf' r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13):
gres14 gf' r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13.
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 grespectful14_iff T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 T13 (gf gf': rel14 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 T13 -> rel14 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 T13) r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13
(EQ: forall r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13, gf r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 <-> gf' r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13):
gres14 gf r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 <-> gres14 gf' r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13.
Proof.
split; intros.
- eapply grespectful14_impl; [apply H | apply EQ].
- eapply grespectful14_impl; [apply H | split; apply EQ].
Qed.
Hint Constructors sound14.
Hint Constructors respectful14.
Hint Constructors gres14.
Hint Resolve gfclo14_mon : paco.
Hint Resolve gfgres14_mon : paco.
Hint Resolve grespectful14_incl.
Hint Resolve rclo14_mon: paco.
Hint Constructors weak_respectful14.
Ltac pupto14_init := eapply upto14_init; [eauto with paco|].
Ltac pupto14_final := first [eapply upto14_final; [eauto with paco|] | eapply grespectful14_incl].
Ltac pupto14 H := first [eapply upto14_step|eapply upto14_step_under]; [|eapply H|]; [eauto with paco|].