fact stringlengths 6 5.47k | type stringclasses 15
values | library stringclasses 1
value | imports listlengths 0 4 | filename stringclasses 39
values | symbolic_name stringlengths 2 26 | docstring stringclasses 1
value |
|---|---|---|---|---|---|---|
rel := (rel10 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9). | Local Notation | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | rel | |
RClo . | Section | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | RClo | |
rclo10 (clo: rel->rel) (r: rel): rel :=
| rclo10_base
x0 x1 x2 x3 x4 x5 x6 x7 x8 x9
(IN: r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9):
@rclo10 clo r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9
| rclo10_clo'
r' x0 x1 x2 x3 x4 x5 x6 x7 x8 x9
(LE: r' <10= rclo10 clo r)
(IN: clo r' x0 x1 x2 x3 x4 x5 x6 x7 x8 x9):
@rclo10... | Inductive | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | rclo10 | |
rclo10_mon_gen clo clo' r r' x0 x1 x2 x3 x4 x5 x6 x7 x8 x9
(IN: @rclo10 clo r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9)
(LEclo: clo <11= clo')
(LEr: r <10= r') :
@rclo10 clo' r' x0 x1 x2 x3 x4 x5 x6 x7 x8 x9.
Proof.
induction IN; intros.
- econstructor 1. apply LEr, IN.
- econstructor 2; [intros; eapply ... | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | rclo10_mon_gen | |
rclo10_mon clo:
monotone10 (rclo10 clo).
Proof.
repeat intro. eapply rclo10_mon_gen; [apply IN|intros; apply PR|apply LE].
Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | rclo10_mon | |
rclo10_clo clo r:
clo (rclo10 clo r) <10= rclo10 clo r.
Proof.
intros. econstructor 2; [|apply PR].
intros. apply PR0.
Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | rclo10_clo | |
rclo10_clo_base clo r:
clo r <10= rclo10 clo r.
Proof.
intros. eapply rclo10_clo', PR.
intros. apply rclo10_base, PR0.
Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | rclo10_clo_base | |
rclo10_rclo clo r:
rclo10 clo (rclo10 clo r) <10= rclo10 clo r.
Proof.
intros. induction PR.
- eapply IN.
- econstructor 2; [eapply H | eapply IN].
Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | rclo10_rclo | |
rclo10_compose clo r:
rclo10 (rclo10 clo) r <10= rclo10 clo r.
Proof.
intros. induction PR.
- apply rclo10_base, IN.
- apply rclo10_rclo.
eapply rclo10_mon; [apply IN|apply H].
Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | rclo10_compose | |
Main .
Variable gf: rel -> rel.
Hypothesis gf_mon: monotone10 gf. | Section | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | Main | |
gpaco10 clo r rg x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 : Prop :=
| gpaco10_intro (IN: @rclo10 clo (paco10 (compose gf (rclo10 clo)) (rg \10/ r) \10/ r) x0 x1 x2 x3 x4 x5 x6 x7 x8 x9)
. | Variant | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | gpaco10 | |
gupaco10 clo r := gpaco10 clo r r. | Definition | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | gupaco10 | |
gpaco10_def_mon clo : monotone10 (compose gf (rclo10 clo)).
Proof.
eapply monotone10_compose. apply gf_mon. apply rclo10_mon.
Qed.
#[local] Hint Resolve gpaco10_def_mon : paco. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | gpaco10_def_mon | |
gpaco10_mon clo r r' rg rg' x0 x1 x2 x3 x4 x5 x6 x7 x8 x9
(IN: @gpaco10 clo r rg x0 x1 x2 x3 x4 x5 x6 x7 x8 x9)
(LEr: r <10= r')
(LErg: rg <10= rg'):
@gpaco10 clo r' rg' x0 x1 x2 x3 x4 x5 x6 x7 x8 x9.
Proof.
destruct IN. econstructor.
eapply rclo10_mon. apply IN.
intros. destruct PR; [|right; ... | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | gpaco10_mon | |
gupaco10_mon clo r r' x0 x1 x2 x3 x4 x5 x6 x7 x8 x9
(IN: @gupaco10 clo r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9)
(LEr: r <10= r'):
@gupaco10 clo r' x0 x1 x2 x3 x4 x5 x6 x7 x8 x9.
Proof.
eapply gpaco10_mon. apply IN. apply LEr. apply LEr.
Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | gupaco10_mon | |
gpaco10_base clo r rg: r <10= gpaco10 clo r rg.
Proof.
econstructor. apply rclo10_base. right. apply PR.
Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | gpaco10_base | |
gpaco10_gen_guard clo r rg:
gpaco10 clo r (rg \10/ r) <10= gpaco10 clo r rg.
Proof.
intros. destruct PR. econstructor.
eapply rclo10_mon. apply IN. intros.
destruct PR; [|right; apply H].
left. eapply paco10_mon_gen; intros. apply H. apply PR.
destruct PR. apply H0. right. apply H0.
Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | gpaco10_gen_guard | |
gpaco10_rclo clo r rg:
rclo10 clo r <10= gpaco10 clo r rg.
Proof.
intros. econstructor.
eapply rclo10_mon. apply PR.
intros. right. apply PR0.
Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | gpaco10_rclo | |
gpaco10_clo clo r rg:
clo r <10= gpaco10 clo r rg.
Proof.
intros. apply gpaco10_rclo. eapply rclo10_clo_base, PR.
Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | gpaco10_clo | |
gpaco10_gen_rclo clo r rg:
gpaco10 (rclo10 clo) r rg <10= gpaco10 clo r rg.
Proof.
intros. destruct PR. econstructor.
apply rclo10_compose.
eapply rclo10_mon. apply IN. intros.
destruct PR; [|right; apply H].
left. eapply paco10_mon_gen; intros; [apply H| |apply PR].
eapply gf_mon, rclo10_compose. apply P... | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | gpaco10_gen_rclo | |
gpaco10_step_gen clo r rg:
gf (gpaco10 clo (rg \10/ r) (rg \10/ r)) <10= gpaco10 clo r rg.
Proof.
intros. econstructor. apply rclo10_base. left.
pstep. eapply gf_mon. apply PR.
intros. destruct PR0. eapply rclo10_mon. apply IN.
intros. destruct PR0.
- left. eapply paco10_mon. apply H. intros. destruct PR0; ... | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | gpaco10_step_gen | |
gpaco10_step clo r rg:
gf (gpaco10 clo rg rg) <10= gpaco10 clo r rg.
Proof.
intros. apply gpaco10_step_gen.
eapply gf_mon. apply PR. intros.
eapply gpaco10_mon. apply PR0. left; apply PR1. left; apply PR1.
Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | gpaco10_step | |
gpaco10_final clo r rg:
(r \10/ paco10 gf rg) <10= gpaco10 clo r rg.
Proof.
intros. destruct PR. apply gpaco10_base, H.
econstructor. apply rclo10_base.
left. eapply paco10_mon_gen. apply H.
- intros. eapply gf_mon. apply PR.
intros. apply rclo10_base. apply PR0.
- intros. left. apply PR.
Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | gpaco10_final | |
gpaco10_unfold clo r rg:
gpaco10 clo r rg <10= rclo10 clo (gf (gupaco10 clo (rg \10/ r)) \10/ r).
Proof.
intros. destruct PR.
eapply rclo10_mon. apply IN.
intros. destruct PR; cycle 1. right; apply H.
left. _punfold H; [|apply gpaco10_def_mon].
eapply gf_mon. apply H.
intros. econstructor.
eapply rclo10... | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | gpaco10_unfold | |
gpaco10_cofix clo r rg
l (OBG: forall rr (INC: rg <10= rr) (CIH: l <10= rr), l <10= gpaco10 clo r rr):
l <10= gpaco10 clo r rg.
Proof.
assert (IN: l <10= gpaco10 clo r (rg \10/ l)).
{ intros. apply OBG; [left; apply PR0 | right; apply PR0 | apply PR]. }
clear OBG. intros. apply IN in PR.
destruct PR. e... | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | gpaco10_cofix | |
gpaco10_gupaco clo r rg:
gupaco10 clo (gpaco10 clo r rg) <10= gpaco10 clo r rg.
Proof.
eapply gpaco10_cofix.
intros. destruct PR. econstructor.
apply rclo10_rclo. eapply rclo10_mon. apply IN.
intros. destruct PR.
- apply rclo10_base. left.
eapply paco10_mon. apply H.
intros. left; apply CIH.
eco... | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | gpaco10_gupaco | |
gpaco10_gpaco clo r rg:
gpaco10 clo (gpaco10 clo r rg) (gupaco10 clo (rg \10/ r)) <10= gpaco10 clo r rg.
Proof.
intros. apply gpaco10_unfold in PR.
econstructor. apply rclo10_rclo. eapply rclo10_mon. apply PR. clear x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 PR. intros.
destruct PR; [|destruct H; apply IN].
apply rclo10_b... | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | gpaco10_gpaco | |
gpaco10_uclo uclo clo r rg
(LEclo: uclo <11= gupaco10 clo) :
uclo (gpaco10 clo r rg) <10= gpaco10 clo r rg.
Proof.
intros. apply gpaco10_gupaco. apply LEclo, PR.
Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | gpaco10_uclo | |
gpaco10_weaken clo r rg:
gpaco10 (gupaco10 clo) r rg <10= gpaco10 clo r rg.
Proof.
intros. apply gpaco10_unfold in PR.
induction PR.
- destruct IN; cycle 1. apply gpaco10_base, H.
apply gpaco10_step_gen. eapply gf_mon. apply H.
clear x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 H.
eapply gpaco10_cofix. intros.
... | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | gpaco10_weaken | |
GeneralMonotonicity .
Variable gf: rel -> rel. | Section | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | GeneralMonotonicity | |
gpaco10_mon_gen (gf' clo clo': rel -> rel) x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 r r' rg rg'
(IN: @gpaco10 gf clo r rg x0 x1 x2 x3 x4 x5 x6 x7 x8 x9)
(gf_mon: monotone10 gf)
(LEgf: gf <11= gf')
(LEclo: clo <11= clo')
(LEr: r <10= r')
(LErg: rg <10= rg') :
@gpaco10 gf' clo' r' rg' x0 x1 x2 ... | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | gpaco10_mon_gen | |
gpaco10_mon_bot (gf' clo clo': rel -> rel) x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 r' rg'
(IN: @gpaco10 gf clo bot10 bot10 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9)
(gf_mon: monotone10 gf)
(LEgf: gf <11= gf')
(LEclo: clo <11= clo'):
@gpaco10 gf' clo' r' rg' x0 x1 x2 x3 x4 x5 x6 x7 x8 x9.
Proof.
eapply gpaco10_mo... | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | gpaco10_mon_bot | |
gupaco10_mon_gen (gf' clo clo': rel -> rel) x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 r r'
(IN: @gupaco10 gf clo r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9)
(gf_mon: monotone10 gf)
(LEgf: gf <11= gf')
(LEclo: clo <11= clo')
(LEr: r <10= r'):
@gupaco10 gf' clo' r' x0 x1 x2 x3 x4 x5 x6 x7 x8 x9.
Proof.
eapply ... | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | gupaco10_mon_gen | |
Compatibility .
Variable gf: rel -> rel.
Hypothesis gf_mon: monotone10 gf.
Structure compatible10 (clo: rel -> rel) : Prop :=
compat10_intro {
compat10_mon: monotone10 clo;
compat10_compat : forall r,
clo (gf r) <10= gf (clo r);
}.
Structure wcompatible10 clo : Prop :=
wcompat10_intro {... | Section | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | Compatibility | |
rclo10_dist clo
(MON: monotone10 clo)
(DIST: forall r1 r2, clo (r1 \10/ r2) <10= (clo r1 \10/ clo r2)):
forall r1 r2, rclo10 clo (r1 \10/ r2) <10= (rclo10 clo r1 \10/ rclo10 clo r2).
Proof.
intros. induction PR.
+ destruct IN; [left|right]; apply rclo10_base, H.
+ assert (REL: clo (rclo10 clo r1 \10... | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | rclo10_dist | |
rclo10_compat clo
(COM: compatible10 clo):
compatible10 (rclo10 clo).
Proof.
econstructor.
- apply rclo10_mon.
- intros. induction PR.
+ eapply gf_mon. apply IN.
intros. eapply rclo10_base. apply PR.
+ eapply gf_mon.
* eapply COM. eapply COM. apply IN. apply H.
* intros. eapply r... | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | rclo10_compat | |
rclo10_wcompat clo
(COM: wcompatible10 clo):
wcompatible10 (rclo10 clo).
Proof.
econstructor.
- apply rclo10_mon.
- intros. induction PR.
+ eapply gf_mon. apply IN.
intros. apply gpaco10_base. apply PR.
+ eapply gf_mon.
* eapply COM. eapply COM. apply IN. apply H.
* intros. eappl... | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | rclo10_wcompat | |
compat10_wcompat clo
(CMP: compatible10 clo):
wcompatible10 clo.
Proof.
econstructor. apply CMP.
intros. apply CMP in PR.
eapply gf_mon. apply PR.
intros. apply gpaco10_clo, PR0.
Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | compat10_wcompat | |
wcompat10_compat clo
(WCMP: wcompatible10 clo) :
compatible10 (gupaco10 gf clo).
Proof.
econstructor.
{ red; intros. eapply gpaco10_mon. apply IN. apply LE. apply LE. }
intros. apply gpaco10_unfold in PR; [|apply gf_mon].
induction PR.
- destruct IN; cycle 1.
+ eapply gf_mon. apply H.
intro... | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | wcompat10_compat | |
wcompat10_union clo1 clo2
(WCMP1: wcompatible10 clo1)
(WCMP2: wcompatible10 clo2):
wcompatible10 (clo1 \11/ clo2).
Proof.
econstructor.
- apply monotone10_union. apply WCMP1. apply WCMP2.
- intros. destruct PR.
+ apply WCMP1 in H. eapply gf_mon. apply H.
intros. eapply gupaco10_mon_gen. ap... | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | wcompat10_union | |
Soundness .
Variable gf: rel -> rel.
Hypothesis gf_mon: monotone10 gf. | Section | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | Soundness | |
gpaco10_compat_init clo
(CMP: compatible10 gf clo):
gpaco10 gf clo bot10 bot10 <10= paco10 gf bot10.
Proof.
intros. destruct PR. revert x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 IN.
pcofix CIH. intros.
pstep. eapply gf_mon; [| right; apply CIH, rclo10_rclo, PR].
apply compat10_compat with (gf:=gf). apply rclo10_co... | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | gpaco10_compat_init | |
gpaco10_init clo
(WCMP: wcompatible10 gf clo):
gpaco10 gf clo bot10 bot10 <10= paco10 gf bot10.
Proof.
intros. eapply gpaco10_compat_init.
- apply wcompat10_compat, WCMP. apply gf_mon.
- eapply gpaco10_mon_bot. apply PR. apply gf_mon. intros; apply PR0.
intros. apply gpaco10_clo, PR0.
Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | gpaco10_init | |
gpaco10_unfold_bot clo
(WCMP: wcompatible10 gf clo):
gpaco10 gf clo bot10 bot10 <10= gf (gpaco10 gf clo bot10 bot10).
Proof.
intros. apply gpaco10_init in PR; [|apply WCMP].
_punfold PR; [..|apply gf_mon].
eapply gf_mon. apply PR.
intros. destruct PR0; [|contradiction]. apply gpaco10_final. apply gf_mon... | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | gpaco10_unfold_bot | |
Distributivity .
Variable gf: rel -> rel.
Hypothesis gf_mon: monotone10 gf. | Section | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | Distributivity | |
gpaco10_dist clo r rg
(CMP: wcompatible10 gf clo)
(DIST: forall r1 r2, clo (r1 \10/ r2) <10= (clo r1 \10/ clo r2)):
gpaco10 gf clo r rg <10= (paco10 gf (rclo10 clo (rg \10/ r)) \10/ rclo10 clo r).
Proof.
intros. apply gpaco10_unfold in PR; [|apply gf_mon].
apply rclo10_dist in PR; [|apply CMP|apply DI... | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | gpaco10_dist | |
gpaco10_dist_reverse clo r rg:
(paco10 gf (rclo10 clo (rg \10/ r)) \10/ rclo10 clo r) <10= gpaco10 gf clo r rg.
Proof.
intros. destruct PR; cycle 1.
- eapply gpaco10_rclo. apply H.
- econstructor. apply rclo10_base. left.
revert x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 H. pcofix CIH; intros.
_punfold H0; [|apply g... | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | gpaco10_dist_reverse | |
Companion .
Variable gf: rel -> rel.
Hypothesis gf_mon: monotone10 gf. | Section | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | Companion | |
cpn10 (r: rel) x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 : Prop :=
| cpn10_intro
clo
(COM: compatible10 gf clo)
(CLO: clo r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9)
. | Inductive | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | cpn10 | |
cpn10_mon : monotone10 cpn10.
Proof.
red. intros.
destruct IN. exists clo.
- apply COM.
- eapply compat10_mon; [apply COM|apply CLO|apply LE].
Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | cpn10_mon | |
cpn10_greatest : forall clo (COM: compatible10 gf clo), clo <11= cpn10.
Proof. intros. econstructor;[apply COM|apply PR]. Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | cpn10_greatest | |
cpn10_compat : compatible10 gf cpn10.
Proof.
econstructor; [apply cpn10_mon|intros].
destruct PR; eapply gf_mon with (r:=clo r).
- eapply (compat10_compat COM); apply CLO.
- intros. econstructor; [apply COM|apply PR].
Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | cpn10_compat | |
cpn10_wcompat : wcompatible10 gf cpn10.
Proof. apply compat10_wcompat, cpn10_compat. apply gf_mon. Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | cpn10_wcompat | |
cpn10_gupaco :
gupaco10 gf cpn10 <11= cpn10.
Proof.
intros. eapply cpn10_greatest, PR. apply wcompat10_compat. apply gf_mon. apply cpn10_wcompat.
Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | cpn10_gupaco | |
cpn10_cpn r:
cpn10 (cpn10 r) <10= cpn10 r.
Proof.
intros. apply cpn10_gupaco, gpaco10_gupaco, gpaco10_clo. apply gf_mon.
eapply cpn10_mon, gpaco10_clo. apply PR.
Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | cpn10_cpn | |
cpn10_base r:
r <10= cpn10 r.
Proof.
intros. apply cpn10_gupaco. apply gpaco10_base, PR.
Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | cpn10_base | |
cpn10_clo r clo (LE: clo <11= cpn10):
clo (cpn10 r) <10= cpn10 r.
Proof.
intros. apply cpn10_cpn, LE, PR.
Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | cpn10_clo | |
cpn10_step r:
gf (cpn10 r) <10= cpn10 r.
Proof.
intros. apply cpn10_gupaco. apply gpaco10_step. apply gf_mon.
eapply gf_mon, gpaco10_clo. apply PR.
Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | cpn10_step | |
cpn10_uclo uclo
(MON: monotone10 uclo)
(WCOM: forall r, uclo (gf r) <10= gf (gupaco10 gf (uclo \11/ cpn10) r)):
uclo <11= gupaco10 gf cpn10.
Proof.
intros. apply gpaco10_clo.
exists (gupaco10 gf (uclo \11/ cpn10)).
- apply wcompat10_compat. apply gf_mon.
econstructor.
+ apply monotone10_unio... | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | cpn10_uclo | |
Respectful .
Variable gf: rel -> rel.
Hypothesis gf_mon: monotone10 gf.
Structure wrespectful10 (clo: rel -> rel) : Prop :=
wrespect10_intro {
wrespect10_mon: monotone10 clo;
wrespect10_respect :
forall l r
(LE: l <10= r)
(GF: l <10= gf r),
clo l <10= gf (rc... | Section | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | Respectful | |
gf'10 := id /11\ gf. | Definition | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | gf'10 | |
compatible'10 := compatible10 gf'10. | Definition | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | compatible'10 | |
wrespect10_compatible' clo (RES: wrespectful10 clo):
compatible'10 (rclo10 clo).
Proof.
intros. econstructor. apply rclo10_mon.
intros. destruct RES. split.
{ eapply rclo10_mon. apply PR. intros. apply PR0. }
induction PR; intros.
- eapply gf_mon. apply IN.
intros. apply rclo10_base, PR.
- eapply gf_m... | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | wrespect10_compatible' | |
prespect10_compatible' clo (RES: prespectful10 clo):
compatible'10 (fun r => upaco10 gf (r \10/ clo r)).
Proof.
econstructor.
{ red; intros. eapply upaco10_mon. apply IN.
intros. destruct PR.
- left. apply LE, H.
- right. eapply RES. apply H. intros. apply LE, PR. }
intros r.
assert (LEM: (gf'10 ... | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | prespect10_compatible' | |
grespect10_compatible' clo (RES: grespectful10 clo):
compatible'10 (rclo10 (clo \11/ cpn10 gf)).
Proof.
apply wrespect10_compatible'.
econstructor.
{ red; intros. destruct IN.
- left. eapply RES; [apply H|]. apply LE.
- right. eapply cpn10_mon; [apply H|]. apply LE. }
intros. destruct PR.
- eapply R... | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | grespect10_compatible' | |
compat10_compatible' clo (COM: compatible10 gf clo):
compatible'10 clo.
Proof.
destruct COM. econstructor; [apply compat10_mon0|].
intros. split.
- eapply compat10_mon0; intros; [apply PR| apply PR0].
- apply compat10_compat0.
eapply compat10_mon0; intros; [apply PR| apply PR0].
Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | compat10_compatible' | |
compatible'10_companion clo (RES: compatible'10 clo):
clo <11= cpn10 gf.
Proof.
assert (MON: monotone10 gf'10).
{ econstructor. apply LE, IN.
eapply gf_mon, LE. apply IN.
}
assert (CPN: clo <11= cpn10 gf'10).
{ intros. econstructor. apply RES. apply PR.
}
intros. apply CPN in PR.
econstructor; [|a... | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | compatible'10_companion | |
wrespect10_companion clo (RES: wrespectful10 clo):
clo <11= cpn10 gf.
Proof.
intros. eapply wrespect10_compatible' in RES.
eapply (@compatible'10_companion (rclo10 clo)) in RES; [apply RES|].
eapply rclo10_clo_base, PR.
Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | wrespect10_companion | |
prespect10_companion clo (RES: prespectful10 clo):
clo <11= cpn10 gf.
Proof.
intros. eapply compatible'10_companion. apply prespect10_compatible'. apply RES.
right. right. apply PR.
Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | prespect10_companion | |
grespect10_companion clo (RES: grespectful10 clo):
clo <11= cpn10 gf.
Proof.
intros. eapply grespect10_compatible' in RES.
eapply (@compatible'10_companion (rclo10 (clo \11/ cpn10 gf))); [apply RES|].
apply rclo10_clo_base. left. apply PR.
Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | grespect10_companion | |
wrespect10_uclo clo (RES: wrespectful10 clo):
clo <11= gupaco10 gf (cpn10 gf).
Proof.
intros. eapply gpaco10_clo, wrespect10_companion, PR. apply RES.
Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | wrespect10_uclo | |
prespect10_uclo clo (RES: prespectful10 clo):
clo <11= gupaco10 gf (cpn10 gf).
Proof.
intros. eapply gpaco10_clo, prespect10_companion, PR. apply RES.
Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | prespect10_uclo | |
grespect10_uclo clo (RES: grespectful10 clo):
clo <11= gupaco10 gf (cpn10 gf).
Proof.
intros. eapply gpaco10_clo, grespect10_companion, PR. apply RES.
Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco10 pacotac."
] | gpaco10.v | grespect10_uclo | |
GeneralizedPaco11 .
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: @... | Section | root | [
"Require Export Program.",
"From Paco Require Import paco11 pacotac."
] | gpaco11.v | GeneralizedPaco11 | |
rel := (rel11 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10). | Local Notation | root | [
"Require Export Program.",
"From Paco Require Import paco11 pacotac."
] | gpaco11.v | rel | |
RClo . | Section | root | [
"Require Export Program.",
"From Paco Require Import paco11 pacotac."
] | gpaco11.v | RClo | |
rclo11 (clo: rel->rel) (r: rel): rel :=
| rclo11_base
x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10
(IN: r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10):
@rclo11 clo r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10
| rclo11_clo'
r' x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10
(LE: r' <11= rclo11 clo r)
(IN: clo r' x0 x1 x2 x3 x4 x5 x6 x7 x8 ... | Inductive | root | [
"Require Export Program.",
"From Paco Require Import paco11 pacotac."
] | gpaco11.v | rclo11 | |
rclo11_mon_gen clo clo' r r' x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10
(IN: @rclo11 clo r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10)
(LEclo: clo <12= clo')
(LEr: r <11= r') :
@rclo11 clo' r' x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10.
Proof.
induction IN; intros.
- econstructor 1. apply LEr, IN.
- econstructor 2; [int... | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco11 pacotac."
] | gpaco11.v | rclo11_mon_gen | |
rclo11_mon clo:
monotone11 (rclo11 clo).
Proof.
repeat intro. eapply rclo11_mon_gen; [apply IN|intros; apply PR|apply LE].
Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco11 pacotac."
] | gpaco11.v | rclo11_mon | |
rclo11_clo clo r:
clo (rclo11 clo r) <11= rclo11 clo r.
Proof.
intros. econstructor 2; [|apply PR].
intros. apply PR0.
Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco11 pacotac."
] | gpaco11.v | rclo11_clo | |
rclo11_clo_base clo r:
clo r <11= rclo11 clo r.
Proof.
intros. eapply rclo11_clo', PR.
intros. apply rclo11_base, PR0.
Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco11 pacotac."
] | gpaco11.v | rclo11_clo_base | |
rclo11_rclo clo r:
rclo11 clo (rclo11 clo r) <11= rclo11 clo r.
Proof.
intros. induction PR.
- eapply IN.
- econstructor 2; [eapply H | eapply IN].
Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco11 pacotac."
] | gpaco11.v | rclo11_rclo | |
rclo11_compose clo r:
rclo11 (rclo11 clo) r <11= rclo11 clo r.
Proof.
intros. induction PR.
- apply rclo11_base, IN.
- apply rclo11_rclo.
eapply rclo11_mon; [apply IN|apply H].
Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco11 pacotac."
] | gpaco11.v | rclo11_compose | |
Main .
Variable gf: rel -> rel.
Hypothesis gf_mon: monotone11 gf. | Section | root | [
"Require Export Program.",
"From Paco Require Import paco11 pacotac."
] | gpaco11.v | Main | |
gpaco11 clo r rg x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 : Prop :=
| gpaco11_intro (IN: @rclo11 clo (paco11 (compose gf (rclo11 clo)) (rg \11/ r) \11/ r) x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10)
. | Variant | root | [
"Require Export Program.",
"From Paco Require Import paco11 pacotac."
] | gpaco11.v | gpaco11 | |
gupaco11 clo r := gpaco11 clo r r. | Definition | root | [
"Require Export Program.",
"From Paco Require Import paco11 pacotac."
] | gpaco11.v | gupaco11 | |
gpaco11_def_mon clo : monotone11 (compose gf (rclo11 clo)).
Proof.
eapply monotone11_compose. apply gf_mon. apply rclo11_mon.
Qed.
#[local] Hint Resolve gpaco11_def_mon : paco. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco11 pacotac."
] | gpaco11.v | gpaco11_def_mon | |
gpaco11_mon clo r r' rg rg' x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10
(IN: @gpaco11 clo r rg x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10)
(LEr: r <11= r')
(LErg: rg <11= rg'):
@gpaco11 clo r' rg' x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10.
Proof.
destruct IN. econstructor.
eapply rclo11_mon. apply IN.
intros. destruct P... | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco11 pacotac."
] | gpaco11.v | gpaco11_mon | |
gupaco11_mon clo r r' x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10
(IN: @gupaco11 clo r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10)
(LEr: r <11= r'):
@gupaco11 clo r' x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10.
Proof.
eapply gpaco11_mon. apply IN. apply LEr. apply LEr.
Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco11 pacotac."
] | gpaco11.v | gupaco11_mon | |
gpaco11_base clo r rg: r <11= gpaco11 clo r rg.
Proof.
econstructor. apply rclo11_base. right. apply PR.
Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco11 pacotac."
] | gpaco11.v | gpaco11_base | |
gpaco11_gen_guard clo r rg:
gpaco11 clo r (rg \11/ r) <11= gpaco11 clo r rg.
Proof.
intros. destruct PR. econstructor.
eapply rclo11_mon. apply IN. intros.
destruct PR; [|right; apply H].
left. eapply paco11_mon_gen; intros. apply H. apply PR.
destruct PR. apply H0. right. apply H0.
Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco11 pacotac."
] | gpaco11.v | gpaco11_gen_guard | |
gpaco11_rclo clo r rg:
rclo11 clo r <11= gpaco11 clo r rg.
Proof.
intros. econstructor.
eapply rclo11_mon. apply PR.
intros. right. apply PR0.
Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco11 pacotac."
] | gpaco11.v | gpaco11_rclo | |
gpaco11_clo clo r rg:
clo r <11= gpaco11 clo r rg.
Proof.
intros. apply gpaco11_rclo. eapply rclo11_clo_base, PR.
Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco11 pacotac."
] | gpaco11.v | gpaco11_clo | |
gpaco11_gen_rclo clo r rg:
gpaco11 (rclo11 clo) r rg <11= gpaco11 clo r rg.
Proof.
intros. destruct PR. econstructor.
apply rclo11_compose.
eapply rclo11_mon. apply IN. intros.
destruct PR; [|right; apply H].
left. eapply paco11_mon_gen; intros; [apply H| |apply PR].
eapply gf_mon, rclo11_compose. apply P... | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco11 pacotac."
] | gpaco11.v | gpaco11_gen_rclo | |
gpaco11_step_gen clo r rg:
gf (gpaco11 clo (rg \11/ r) (rg \11/ r)) <11= gpaco11 clo r rg.
Proof.
intros. econstructor. apply rclo11_base. left.
pstep. eapply gf_mon. apply PR.
intros. destruct PR0. eapply rclo11_mon. apply IN.
intros. destruct PR0.
- left. eapply paco11_mon. apply H. intros. destruct PR0; ... | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco11 pacotac."
] | gpaco11.v | gpaco11_step_gen | |
gpaco11_step clo r rg:
gf (gpaco11 clo rg rg) <11= gpaco11 clo r rg.
Proof.
intros. apply gpaco11_step_gen.
eapply gf_mon. apply PR. intros.
eapply gpaco11_mon. apply PR0. left; apply PR1. left; apply PR1.
Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco11 pacotac."
] | gpaco11.v | gpaco11_step | |
gpaco11_final clo r rg:
(r \11/ paco11 gf rg) <11= gpaco11 clo r rg.
Proof.
intros. destruct PR. apply gpaco11_base, H.
econstructor. apply rclo11_base.
left. eapply paco11_mon_gen. apply H.
- intros. eapply gf_mon. apply PR.
intros. apply rclo11_base. apply PR0.
- intros. left. apply PR.
Qed. | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco11 pacotac."
] | gpaco11.v | gpaco11_final | |
gpaco11_unfold clo r rg:
gpaco11 clo r rg <11= rclo11 clo (gf (gupaco11 clo (rg \11/ r)) \11/ r).
Proof.
intros. destruct PR.
eapply rclo11_mon. apply IN.
intros. destruct PR; cycle 1. right; apply H.
left. _punfold H; [|apply gpaco11_def_mon].
eapply gf_mon. apply H.
intros. econstructor.
eapply rclo11... | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco11 pacotac."
] | gpaco11.v | gpaco11_unfold | |
gpaco11_cofix clo r rg
l (OBG: forall rr (INC: rg <11= rr) (CIH: l <11= rr), l <11= gpaco11 clo r rr):
l <11= gpaco11 clo r rg.
Proof.
assert (IN: l <11= gpaco11 clo r (rg \11/ l)).
{ intros. apply OBG; [left; apply PR0 | right; apply PR0 | apply PR]. }
clear OBG. intros. apply IN in PR.
destruct PR. e... | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco11 pacotac."
] | gpaco11.v | gpaco11_cofix | |
gpaco11_gupaco clo r rg:
gupaco11 clo (gpaco11 clo r rg) <11= gpaco11 clo r rg.
Proof.
eapply gpaco11_cofix.
intros. destruct PR. econstructor.
apply rclo11_rclo. eapply rclo11_mon. apply IN.
intros. destruct PR.
- apply rclo11_base. left.
eapply paco11_mon. apply H.
intros. left; apply CIH.
eco... | Lemma | root | [
"Require Export Program.",
"From Paco Require Import paco11 pacotac."
] | gpaco11.v | gpaco11_gupaco |
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