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| type
stringclasses 15
values | library
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value | imports
listlengths 0
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| filename
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values | symbolic_name
stringlengths 2
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|---|---|---|---|---|---|---|
stream (A:Type) :=
| snil : stream A
| scons : A -> stream A -> stream A
.
Arguments scons {_} _ _.
|
CoInductive
|
root
|
[
"Require Import ZArith List String.",
"Require Import Paco.",
"Require Import Program Classical."
] |
examples.v
|
stream
| |
id_match_stream {A} (s:stream A) : stream A :=
match s with
| snil => snil
| scons x t => scons x t
end.
|
Definition
|
root
|
[
"Require Import ZArith List String.",
"Require Import Paco.",
"Require Import Program Classical."
] |
examples.v
|
id_match_stream
| |
id_stream_eq : forall A (s:stream A), s = id_match_stream s.
Proof.
intros.
destruct s; auto.
Qed.
(* A more relaxed notion of equivalence where the 0's can be inserted finitely often in either
stream. *)
|
Lemma
|
root
|
[
"Require Import ZArith List String.",
"Require Import Paco.",
"Require Import Program Classical."
] |
examples.v
|
id_stream_eq
| |
seq_step (seq : stream nat -> stream nat -> Prop) : stream nat -> stream nat -> Prop :=
| seq_nil : seq_step seq snil snil
| seq_cons : forall s1 s2 n (R : seq s1 s2), seq_step seq (scons n s1) (scons n s2)
| seq_cons_z_l : forall s1 s2, seq_step seq s1 s2 -> seq_step seq (scons 0 s1) s2
| seq_cons_z_r : forall s1 s2, seq_step seq s1 s2 -> seq_step seq s1 (scons 0 s2)
.
#[export] Hint Constructors seq_step : core.
|
Inductive
|
root
|
[
"Require Import ZArith List String.",
"Require Import Paco.",
"Require Import Program Classical."
] |
examples.v
|
seq_step
| |
seq_step_mono : monotone2 seq_step.
Proof.
unfold monotone2. intros x0 x1 r r' IN LE.
induction IN; eauto.
Qed.
#[export] Hint Resolve seq_step_mono : paco.
|
Lemma
|
root
|
[
"Require Import ZArith List String.",
"Require Import Paco.",
"Require Import Program Classical."
] |
examples.v
|
seq_step_mono
| |
seq (s t : stream nat) := paco2 seq_step bot2 s t .
#[export] Hint Unfold seq : core.
|
Definition
|
root
|
[
"Require Import ZArith List String.",
"Require Import Paco.",
"Require Import Program Classical."
] |
examples.v
|
seq
| |
seq_step_refl : forall (R:stream nat -> stream nat -> Prop),
(forall d, R d d) -> forall d, seq_step R d d.
Proof.
intros.
destruct d; constructor; auto.
Qed.
|
Lemma
|
root
|
[
"Require Import ZArith List String.",
"Require Import Paco.",
"Require Import Program Classical."
] |
examples.v
|
seq_step_refl
| |
seq_refl : forall s, seq s s.
Proof.
pcofix CIH.
intros s.
pfold.
destruct s; auto.
Qed.
|
Lemma
|
root
|
[
"Require Import ZArith List String.",
"Require Import Paco.",
"Require Import Program Classical."
] |
examples.v
|
seq_refl
| |
seq_symm : forall s1 s2, seq s1 s2 -> seq s2 s1.
Proof.
pcofix CIH.
intros s1 s2 H.
pfold.
punfold H.
induction H; try constructor; auto.
pclearbot. right. apply CIH. punfold R.
Qed.
Require Import Program Classical.
|
Lemma
|
root
|
[
"Require Import ZArith List String.",
"Require Import Paco.",
"Require Import Program Classical."
] |
examples.v
|
seq_symm
| |
zeros_star (P: stream nat -> Prop) : stream nat -> Prop :=
| zs_base t (BASE: P t): zeros_star P t
| zs_step t (LZ: zeros_star P t): zeros_star P (scons 0 t)
.
#[export] Hint Constructors zeros_star : core.
|
Inductive
|
root
|
[
"Require Import ZArith List String.",
"Require Import Paco.",
"Require Import Program Classical."
] |
examples.v
|
zeros_star
| |
zeros_star_mono : monotone1 zeros_star.
Proof.
red. intros. induction IN; eauto.
Qed.
#[export] Hint Resolve zeros_star_mono : core.
|
Lemma
|
root
|
[
"Require Import ZArith List String.",
"Require Import Paco.",
"Require Import Program Classical."
] |
examples.v
|
zeros_star_mono
| |
zeros_one (P: stream nat -> Prop) : stream nat -> Prop :=
| zo_step t (BASE: P t): zeros_one P (scons 0 t)
.
#[export] Hint Constructors zeros_one : core.
|
Inductive
|
root
|
[
"Require Import ZArith List String.",
"Require Import Paco.",
"Require Import Program Classical."
] |
examples.v
|
zeros_one
| |
zeros_one_mono : monotone1 zeros_one.
Proof.
pmonauto.
Qed.
#[export] Hint Resolve zeros_one_mono : paco.
|
Lemma
|
root
|
[
"Require Import ZArith List String.",
"Require Import Paco.",
"Require Import Program Classical."
] |
examples.v
|
zeros_one_mono
| |
infzeros := paco1 zeros_one bot1.
#[export] Hint Unfold infzeros : core.
|
Definition
|
root
|
[
"Require Import ZArith List String.",
"Require Import Paco.",
"Require Import Program Classical."
] |
examples.v
|
infzeros
| |
nonzero : stream nat -> Prop :=
| nz_nil: nonzero snil
| nz_cons n s (NZ: n <> 0): nonzero (scons n s)
.
#[export] Hint Constructors nonzero : core.
|
Inductive
|
root
|
[
"Require Import ZArith List String.",
"Require Import Paco.",
"Require Import Program Classical."
] |
examples.v
|
nonzero
| |
gseq_cons_or_nil (gseq: stream nat -> stream nat -> Prop) : stream nat -> stream nat -> Prop :=
| gseq_nil : gseq_cons_or_nil gseq snil snil
| gseq_cons s1 s2 n (R: gseq s1 s2) (NZ: n <> 0) : gseq_cons_or_nil gseq (scons n s1) (scons n s2)
.
#[export] Hint Constructors gseq_cons_or_nil : core.
|
Inductive
|
root
|
[
"Require Import ZArith List String.",
"Require Import Paco.",
"Require Import Program Classical."
] |
examples.v
|
gseq_cons_or_nil
| |
gseq_step (gseq: stream nat -> stream nat -> Prop) : stream nat -> stream nat -> Prop :=
| gseq_inf s1 s2 (IZ1: infzeros s1) (IZ2: infzeros s2) : gseq_step gseq s1 s2
| gseq_fin s1 s2 t1 t2
(ZS1: zeros_star (fun t => t = s1) t1)
(ZS2: zeros_star (fun t => t = s2) t2)
(R: gseq_cons_or_nil gseq s1 s2)
: gseq_step gseq t1 t2
.
#[export] Hint Constructors gseq_step : core.
|
Inductive
|
root
|
[
"Require Import ZArith List String.",
"Require Import Paco.",
"Require Import Program Classical."
] |
examples.v
|
gseq_step
| |
gseq_step_mono : monotone2 gseq_step.
Proof.
unfold monotone2. intros.
induction IN; eauto.
eapply gseq_fin; eauto.
destruct R; eauto.
Qed.
#[export] Hint Resolve gseq_step_mono : paco.
|
Lemma
|
root
|
[
"Require Import ZArith List String.",
"Require Import Paco.",
"Require Import Program Classical."
] |
examples.v
|
gseq_step_mono
| |
gseq (s t : stream nat) := paco2 gseq_step bot2 s t .
#[export] Hint Unfold gseq : core.
|
Definition
|
root
|
[
"Require Import ZArith List String.",
"Require Import Paco.",
"Require Import Program Classical."
] |
examples.v
|
gseq
| |
nonzero_not_infzeros : forall s
(ZST: zeros_star nonzero s)
(INF: infzeros s),
False.
Proof.
intros. revert INF. induction ZST.
- intro INF. punfold INF. dependent destruction INF.
dependent destruction BASE. intuition.
- intro INF. apply IHZST.
punfold INF. dependent destruction INF. pclearbot. eauto.
Qed.
|
Lemma
|
root
|
[
"Require Import ZArith List String.",
"Require Import Paco.",
"Require Import Program Classical."
] |
examples.v
|
nonzero_not_infzeros
| |
infzeros_or_finzeros : forall s,
infzeros s \/ zeros_star nonzero s.
Proof.
intros. destruct (classic (zeros_star nonzero s)); eauto.
left. revert s H. pcofix CIH.
intros. destruct s.
- exfalso. eauto.
- destruct n; [|exfalso; eauto].
pfold. econstructor. right. eauto.
Qed.
|
Lemma
|
root
|
[
"Require Import ZArith List String.",
"Require Import Paco.",
"Require Import Program Classical."
] |
examples.v
|
infzeros_or_finzeros
| |
seq_infzeros_imply : forall s t
(R: seq s t) (IZ: infzeros s), infzeros t.
Proof.
pcofix CIH. intros.
punfold R. revert IZ. induction R; intros.
- eapply paco1_mon. eauto. intros. inversion PR.
- pfold. punfold IZ. dependent destruction IZ. pclearbot. eauto.
- punfold IZ. dependent destruction IZ. pclearbot. eauto.
- pfold. eauto.
Qed.
|
Lemma
|
root
|
[
"Require Import ZArith List String.",
"Require Import Paco.",
"Require Import Program Classical."
] |
examples.v
|
seq_infzeros_imply
| |
seq_zeros_star_imply : forall s t
(R: seq s t) (IZ: zeros_star nonzero s), zeros_star nonzero t.
Proof.
intros. revert t R. induction IZ; intros.
- punfold R. induction R; pclearbot; eauto.
+ inversion BASE. eauto.
+ inversion BASE. intuition.
- punfold R. remember(scons 0 t). generalize dependent t.
induction R; intros; pclearbot; dependent destruction Heqs; eauto.
Qed.
|
Lemma
|
root
|
[
"Require Import ZArith List String.",
"Require Import Paco.",
"Require Import Program Classical."
] |
examples.v
|
seq_zeros_star_imply
| |
seq_infzeros_or_finzeros : forall s t
(R: seq s t),
(infzeros s /\ infzeros t) \/
(zeros_star nonzero s /\ zeros_star nonzero t).
Proof.
intros. destruct (@infzeros_or_finzeros s).
- eauto using seq_infzeros_imply.
- eauto using seq_zeros_star_imply.
Qed.
|
Lemma
|
root
|
[
"Require Import ZArith List String.",
"Require Import Paco.",
"Require Import Program Classical."
] |
examples.v
|
seq_infzeros_or_finzeros
| |
seq_zero_l : forall s t
(EQ : seq (scons 0 s) t),
seq s t.
Proof.
intros. punfold EQ. pfold.
remember (scons 0 s). generalize dependent s.
induction EQ; intros; dependent destruction Heqs0; pclearbot; eauto.
punfold R.
Qed.
|
Lemma
|
root
|
[
"Require Import ZArith List String.",
"Require Import Paco.",
"Require Import Program Classical."
] |
examples.v
|
seq_zero_l
| |
seq_zero_r : forall s t
(EQ : seq s (scons 0 t)),
seq s t.
Proof.
intros. punfold EQ. pfold.
remember (scons 0 t). generalize dependent t.
induction EQ; intros; dependent destruction Heqs0; pclearbot; eauto.
punfold R.
Qed.
|
Lemma
|
root
|
[
"Require Import ZArith List String.",
"Require Import Paco.",
"Require Import Program Classical."
] |
examples.v
|
seq_zero_r
| |
zero_gseq_l : forall r s t
(R : paco2 gseq_step r s t),
paco2 gseq_step r (scons 0 s) t.
Proof.
intros. punfold R. pfold. destruct R; eauto.
econstructor; eauto. pfold. eauto.
Qed.
|
Lemma
|
root
|
[
"Require Import ZArith List String.",
"Require Import Paco.",
"Require Import Program Classical."
] |
examples.v
|
zero_gseq_l
| |
zero_gseq_r : forall r s t
(R : paco2 gseq_step r s t),
paco2 gseq_step r s (scons 0 t).
Proof.
intros. punfold R. pfold. destruct R; eauto.
econstructor; eauto. pfold. eauto.
Qed.
|
Lemma
|
root
|
[
"Require Import ZArith List String.",
"Require Import Paco.",
"Require Import Program Classical."
] |
examples.v
|
zero_gseq_r
| |
seq_implies_gseq : forall s t
(R: seq s t), gseq s t.
Proof.
pcofix CIH.
intros. destruct (seq_infzeros_or_finzeros R) as [[]|[]]; eauto.
induction H.
- induction H0.
+ pfold. punfold R. dependent destruction R; pclearbot; eauto.
* dependent destruction BASE. eauto 10.
* dependent destruction BASE. intuition.
* dependent destruction BASE0. intuition.
+ eauto using seq_zero_r, zero_gseq_r.
- eauto using seq_zero_l, zero_gseq_l.
Qed.
|
Lemma
|
root
|
[
"Require Import ZArith List String.",
"Require Import Paco.",
"Require Import Program Classical."
] |
examples.v
|
seq_implies_gseq
| |
gseq_implies_seq : forall s t
(R: gseq s t), seq s t.
Proof.
pcofix CIH; intros.
punfold R. destruct R.
- punfold IZ1. punfold IZ2.
dependent destruction IZ1. dependent destruction IZ2. pclearbot.
pfold. econstructor. right. eauto.
- induction ZS1; subst.
+ induction ZS2; subst.
* pfold. dependent destruction R; pclearbot; eauto.
* pfold. punfold IHZS2.
+ pfold. punfold IHZS1.
Qed.
|
Lemma
|
root
|
[
"Require Import ZArith List String.",
"Require Import Paco.",
"Require Import Program Classical."
] |
examples.v
|
gseq_implies_seq
| |
gseq_cons_or_nil_nonzero_l : forall r s t
(R : gseq_cons_or_nil r s t),
nonzero s.
Proof. intros; destruct R; eauto. Qed.
|
Lemma
|
root
|
[
"Require Import ZArith List String.",
"Require Import Paco.",
"Require Import Program Classical."
] |
examples.v
|
gseq_cons_or_nil_nonzero_l
| |
gseq_cons_or_nil_nonzero_r : forall r s t
(R : gseq_cons_or_nil r s t),
nonzero t.
Proof. intros; destruct R; eauto. Qed.
|
Lemma
|
root
|
[
"Require Import ZArith List String.",
"Require Import Paco.",
"Require Import Program Classical."
] |
examples.v
|
gseq_cons_or_nil_nonzero_r
| |
zeros_star_nonzero_uniq : forall s1 s2 t
(ZS1: zeros_star (fun s => s = s1) t)
(ZS2: zeros_star (fun s => s = s2) t)
(NZ1: nonzero s1)
(NZ2: nonzero s2),
s1 = s2.
Proof.
intros s1 s2 t ZS1. revert s2.
induction ZS1; subst; intros.
- induction ZS2; subst; eauto.
inversion NZ1. intuition.
- dependent destruction ZS2; eauto.
inversion NZ2. intuition.
Qed.
|
Lemma
|
root
|
[
"Require Import ZArith List String.",
"Require Import Paco.",
"Require Import Program Classical."
] |
examples.v
|
zeros_star_nonzero_uniq
| |
gseq_trans : forall d1 d2 d3
(EQL: gseq d1 d2) (EQR: gseq d2 d3), gseq d1 d3.
Proof.
pcofix CIH; intros.
punfold EQL. punfold EQR. destruct EQL, EQR; eauto.
- exfalso. eapply nonzero_not_infzeros, IZ2.
eapply zeros_star_mono; eauto.
simpl. intros. subst. destruct R; eauto.
- exfalso. eapply nonzero_not_infzeros, IZ1.
eapply zeros_star_mono; eauto.
simpl. intros. subst. destruct R; eauto.
- eapply zeros_star_nonzero_uniq in ZS2;
eauto using gseq_cons_or_nil_nonzero_l, gseq_cons_or_nil_nonzero_r.
subst. pfold. econstructor 2; eauto.
destruct R; dependent destruction R0; pclearbot; eauto.
Qed.
|
Lemma
|
root
|
[
"Require Import ZArith List String.",
"Require Import Paco.",
"Require Import Program Classical."
] |
examples.v
|
gseq_trans
| |
seq_trans : forall d1 d2 d3
(EQL: seq d1 d2) (EQR: seq d2 d3), seq d1 d3.
Proof.
eauto using gseq_trans, seq_implies_gseq, gseq_implies_seq.
Qed.
(**
Tests for [pclearbot]
**)
|
Lemma
|
root
|
[
"Require Import ZArith List String.",
"Require Import Paco.",
"Require Import Program Classical."
] |
examples.v
|
seq_trans
| |
plcearbot_test1 x y
(H: upaco2 seq_step bot2 x y)
:
True.
Proof.
pclearbot.
eauto.
Qed.
|
Lemma
|
root
|
[
"Require Import ZArith List String.",
"Require Import Paco.",
"Require Import Program Classical."
] |
examples.v
|
plcearbot_test1
| |
plcearbot_test2 (H: forall x y, upaco2 seq_step bot2 x y)
:
True.
Proof.
pclearbot.
eauto.
Qed.
|
Lemma
|
root
|
[
"Require Import ZArith List String.",
"Require Import Paco.",
"Require Import Program Classical."
] |
examples.v
|
plcearbot_test2
| |
inv H := inversion H; subst; clear H.
(*** Coinductive stream ***)
|
Ltac
|
root
|
[
"Require Import Paco.",
"Require Import Program.",
"Require Import RelationClasses.",
"Require Import Morphisms."
] |
example_upto.v
|
inv
| |
stream :=
| snil
| scons (n: nat) (s: stream)
.
|
CoInductive
|
root
|
[
"Require Import Paco.",
"Require Import Program.",
"Require Import RelationClasses.",
"Require Import Morphisms."
] |
example_upto.v
|
stream
| |
match_stream (s: stream) :=
match s with
| snil => snil
| scons n s => scons n s
end.
|
Definition
|
root
|
[
"Require Import Paco.",
"Require Import Program.",
"Require Import RelationClasses.",
"Require Import Morphisms."
] |
example_upto.v
|
match_stream
| |
unfold_stream s
:
s = match_stream s.
Proof.
destruct s; auto.
Qed.
(*** Concat of two streams ***)
|
Lemma
|
root
|
[
"Require Import Paco.",
"Require Import Program.",
"Require Import RelationClasses.",
"Require Import Morphisms."
] |
example_upto.v
|
unfold_stream
| |
match_concat concat (s0 s1: stream): stream :=
match s0 with
| snil => s1
| scons n s0 => scons n (concat s0 s1)
end
.
CoFixpoint concat (s0 s1: stream): stream := match_concat concat s0 s1.
Declare Scope stream_scope.
|
Definition
|
root
|
[
"Require Import Paco.",
"Require Import Program.",
"Require Import RelationClasses.",
"Require Import Morphisms."
] |
example_upto.v
|
match_concat
| |
unfold_concat : forall s0 s1, s0 ++ s1 = match_concat concat s0 s1.
Proof.
intros.
rewrite unfold_stream with (concat s0 s1). simpl.
destruct (match_concat concat s0 s1) eqn:T; reflexivity.
Qed.
(*** Bisimulation between two streams ***)
|
Lemma
|
root
|
[
"Require Import Paco.",
"Require Import Program.",
"Require Import RelationClasses.",
"Require Import Morphisms."
] |
example_upto.v
|
unfold_concat
| |
_sim (sim : stream -> stream -> Prop): stream -> stream -> Prop :=
| SimNil: _sim sim snil snil
| SimCons n sl sr (REL: sim sl sr): _sim sim (scons n sl) (scons n sr)
.
#[export] Hint Constructors _sim : core.
|
Inductive
|
root
|
[
"Require Import Paco.",
"Require Import Program.",
"Require Import RelationClasses.",
"Require Import Morphisms."
] |
example_upto.v
|
_sim
| |
_sim_mon : monotone2 (_sim).
Proof.
intros x0 x1 r r' IN LE. induction IN; auto.
Qed.
#[export] Hint Resolve _sim_mon : paco.
|
Lemma
|
root
|
[
"Require Import Paco.",
"Require Import Program.",
"Require Import RelationClasses.",
"Require Import Morphisms."
] |
example_upto.v
|
_sim_mon
| |
sim (sl sr: stream) := paco2 (_sim) bot2 sl sr.
#[export] Hint Unfold sim : core.
(*** First (failing) attempt without upto technique ***)
|
Definition
|
root
|
[
"Require Import Paco.",
"Require Import Program.",
"Require Import RelationClasses.",
"Require Import Morphisms."
] |
example_upto.v
|
sim
| |
sim_concat s0 s1 t0 t1
(EQ0: sim s0 t0)
(EQ1: sim s1 t1)
:
@sim (concat s0 s1) (concat t0 t1)
.
Proof.
revert_until s0. revert s0.
pcofix CIH. intros. pfold.
punfold EQ0. punfold EQ1.
inv EQ0.
- rewrite ! unfold_concat. cbn.
eapply _sim_mon; eauto. intros. pclearbot. left. eapply paco2_mon; eauto. intros.
red in PR0. contradict PR0.
- rewrite ! unfold_concat. cbn. pclearbot.
econstructor; eauto. left.
inv EQ1.
+ (*** We are stuck here..? ***) admit.
+ pclearbot. pfold.
(*** We are stuck here..? ***) admit.
(*** TODO: give better explanation ***)
Abort.
(*** Second attempt with upto technique ***)
|
Lemma
|
root
|
[
"Require Import Paco.",
"Require Import Program.",
"Require Import RelationClasses.",
"Require Import Morphisms."
] |
example_upto.v
|
sim_concat
| |
prefixC (R : stream -> stream -> Prop): stream -> stream -> Prop :=
| prefixC_intro
s0 s1 t0 t1
(REL: sim s0 t0)
(REL: R s1 t1)
:
prefixC R (concat s0 s1) (concat t0 t1)
.
#[export] Hint Constructors prefixC: core.
|
Inductive
|
root
|
[
"Require Import Paco.",
"Require Import Program.",
"Require Import RelationClasses.",
"Require Import Morphisms."
] |
example_upto.v
|
prefixC
| |
prefixC_spec simC
:
prefixC <3= gupaco2 (_sim) (simC)
.
Proof.
gcofix CIH. intros. destruct PR.
punfold REL. inv REL.
- rewrite ! unfold_concat. cbn. gbase. eauto. (* Note: "eauto with paco" also works *)
- gstep.
rewrite ! unfold_concat. cbn.
econstructor; eauto.
pclearbot. gbase. eauto. (* Note: "eauto with paco" also works *)
Qed.
|
Lemma
|
root
|
[
"Require Import Paco.",
"Require Import Program.",
"Require Import RelationClasses.",
"Require Import Morphisms."
] |
example_upto.v
|
prefixC_spec
| |
sim_concat s0 s1 t0 t1
(EQ0: sim s0 t0)
(EQ1: sim s1 t1)
:
@sim (concat s0 s1) (concat t0 t1)
.
Proof.
intros. ginit. { eapply cpn2_wcompat; pmonauto. } guclo prefixC_spec.
Qed.
|
Lemma
|
root
|
[
"Require Import Paco.",
"Require Import Program.",
"Require Import RelationClasses.",
"Require Import Morphisms."
] |
example_upto.v
|
sim_concat
| |
sim_concat_proper : Proper (sim ==> sim ==> sim) concat.
Proof.
repeat intro. eapply sim_concat; eauto.
Qed.
|
Lemma
|
root
|
[
"Require Import Paco.",
"Require Import Program.",
"Require Import RelationClasses.",
"Require Import Morphisms."
] |
example_upto.v
|
sim_concat_proper
| |
GeneralizedPaco0 .
|
Section
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
GeneralizedPaco0
| |
rel := (rel0).
|
Local Notation
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
rel
| |
RClo .
|
Section
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
RClo
| |
rclo0 (clo: rel->rel) (r: rel): rel :=
| rclo0_base
(IN: r):
@rclo0 clo r
| rclo0_clo'
r'
(LE: r' <0= rclo0 clo r)
(IN: clo r'):
@rclo0 clo r
.
|
Inductive
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
rclo0
| |
rclo0_mon_gen clo clo' r r'
(IN: @rclo0 clo r)
(LEclo: clo <1= clo')
(LEr: r <0= r') :
@rclo0 clo' r'.
Proof.
induction IN; intros.
- econstructor 1. apply LEr, IN.
- econstructor 2; [intros; eapply H, PR|apply LEclo, IN].
Qed.
|
Lemma
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
rclo0_mon_gen
| |
rclo0_mon clo:
monotone0 (rclo0 clo).
Proof.
repeat intro. eapply rclo0_mon_gen; [apply IN|intros; apply PR|apply LE].
Qed.
|
Lemma
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
rclo0_mon
| |
rclo0_clo clo r:
clo (rclo0 clo r) <0= rclo0 clo r.
Proof.
intros. econstructor 2; [|apply PR].
intros. apply PR0.
Qed.
|
Lemma
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
rclo0_clo
| |
rclo0_clo_base clo r:
clo r <0= rclo0 clo r.
Proof.
intros. eapply rclo0_clo', PR.
intros. apply rclo0_base, PR0.
Qed.
|
Lemma
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
rclo0_clo_base
| |
rclo0_rclo clo r:
rclo0 clo (rclo0 clo r) <0= rclo0 clo r.
Proof.
intros. induction PR.
- eapply IN.
- econstructor 2; [eapply H | eapply IN].
Qed.
|
Lemma
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
rclo0_rclo
| |
rclo0_compose clo r:
rclo0 (rclo0 clo) r <0= rclo0 clo r.
Proof.
intros. induction PR.
- apply rclo0_base, IN.
- apply rclo0_rclo.
eapply rclo0_mon; [apply IN|apply H].
Qed.
|
Lemma
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
rclo0_compose
| |
Main .
Variable gf: rel -> rel.
Hypothesis gf_mon: monotone0 gf.
|
Section
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
Main
| |
gpaco0 clo r rg : Prop :=
| gpaco0_intro (IN: @rclo0 clo (paco0 (compose gf (rclo0 clo)) (rg \0/ r) \0/ r))
.
|
Variant
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
gpaco0
| |
gupaco0 clo r := gpaco0 clo r r.
|
Definition
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
gupaco0
| |
gpaco0_def_mon clo : monotone0 (compose gf (rclo0 clo)).
Proof.
eapply monotone0_compose. apply gf_mon. apply rclo0_mon.
Qed.
#[local] Hint Resolve gpaco0_def_mon : paco.
|
Lemma
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
gpaco0_def_mon
| |
gpaco0_mon clo r r' rg rg'
(IN: @gpaco0 clo r rg)
(LEr: r <0= r')
(LErg: rg <0= rg'):
@gpaco0 clo r' rg'.
Proof.
destruct IN. econstructor.
eapply rclo0_mon. apply IN.
intros. destruct PR; [|right; apply LEr, H].
left. eapply paco0_mon. apply H.
intros. destruct PR.
- left. apply LErg, H0.
- right. apply LEr, H0.
Qed.
|
Lemma
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
gpaco0_mon
| |
gupaco0_mon clo r r'
(IN: @gupaco0 clo r)
(LEr: r <0= r'):
@gupaco0 clo r'.
Proof.
eapply gpaco0_mon. apply IN. apply LEr. apply LEr.
Qed.
|
Lemma
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
gupaco0_mon
| |
gpaco0_base clo r rg: r <0= gpaco0 clo r rg.
Proof.
econstructor. apply rclo0_base. right. apply PR.
Qed.
|
Lemma
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
gpaco0_base
| |
gpaco0_gen_guard clo r rg:
gpaco0 clo r (rg \0/ r) <0= gpaco0 clo r rg.
Proof.
intros. destruct PR. econstructor.
eapply rclo0_mon. apply IN. intros.
destruct PR; [|right; apply H].
left. eapply paco0_mon_gen; intros. apply H. apply PR.
destruct PR. apply H0. right. apply H0.
Qed.
|
Lemma
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
gpaco0_gen_guard
| |
gpaco0_rclo clo r rg:
rclo0 clo r <0= gpaco0 clo r rg.
Proof.
intros. econstructor.
eapply rclo0_mon. apply PR.
intros. right. apply PR0.
Qed.
|
Lemma
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
gpaco0_rclo
| |
gpaco0_clo clo r rg:
clo r <0= gpaco0 clo r rg.
Proof.
intros. apply gpaco0_rclo. eapply rclo0_clo_base, PR.
Qed.
|
Lemma
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
gpaco0_clo
| |
gpaco0_gen_rclo clo r rg:
gpaco0 (rclo0 clo) r rg <0= gpaco0 clo r rg.
Proof.
intros. destruct PR. econstructor.
apply rclo0_compose.
eapply rclo0_mon. apply IN. intros.
destruct PR; [|right; apply H].
left. eapply paco0_mon_gen; intros; [apply H| |apply PR].
eapply gf_mon, rclo0_compose. apply PR.
Qed.
|
Lemma
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
gpaco0_gen_rclo
| |
gpaco0_step_gen clo r rg:
gf (gpaco0 clo (rg \0/ r) (rg \0/ r)) <0= gpaco0 clo r rg.
Proof.
intros. econstructor. apply rclo0_base. left.
pstep. eapply gf_mon. apply PR.
intros. destruct PR0. eapply rclo0_mon. apply IN.
intros. destruct PR0.
- left. eapply paco0_mon. apply H. intros. destruct PR0; apply H0.
- right. apply H.
Qed.
|
Lemma
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
gpaco0_step_gen
| |
gpaco0_step clo r rg:
gf (gpaco0 clo rg rg) <0= gpaco0 clo r rg.
Proof.
intros. apply gpaco0_step_gen.
eapply gf_mon. apply PR. intros.
eapply gpaco0_mon. apply PR0. left; apply PR1. left; apply PR1.
Qed.
|
Lemma
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
gpaco0_step
| |
gpaco0_final clo r rg:
(r \0/ paco0 gf rg) <0= gpaco0 clo r rg.
Proof.
intros. destruct PR. apply gpaco0_base, H.
econstructor. apply rclo0_base.
left. eapply paco0_mon_gen. apply H.
- intros. eapply gf_mon. apply PR.
intros. apply rclo0_base. apply PR0.
- intros. left. apply PR.
Qed.
|
Lemma
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
gpaco0_final
| |
gpaco0_unfold clo r rg:
gpaco0 clo r rg <0= rclo0 clo (gf (gupaco0 clo (rg \0/ r)) \0/ r).
Proof.
intros. destruct PR.
eapply rclo0_mon. apply IN.
intros. destruct PR; cycle 1. right; apply H.
left. _punfold H; [|apply gpaco0_def_mon].
eapply gf_mon. apply H.
intros. econstructor.
eapply rclo0_mon. apply PR.
intros. destruct PR0; cycle 1. right. apply H0.
left. eapply paco0_mon. apply H0.
intros. left. apply PR0.
Qed.
|
Lemma
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
gpaco0_unfold
| |
gpaco0_cofix clo r rg
l (OBG: forall rr (INC: rg <0= rr) (CIH: l <0= rr), l <0= gpaco0 clo r rr):
l <0= gpaco0 clo r rg.
Proof.
assert (IN: l <0= gpaco0 clo r (rg \0/ l)).
{ intros. apply OBG; [left; apply PR0 | right; apply PR0 | apply PR]. }
clear OBG. intros. apply IN in PR.
destruct PR. econstructor.
eapply rclo0_mon. apply IN0.
clear IN0.
intros. destruct PR; [|right; apply H].
left. revert H.
pcofix CIH. intros.
_punfold H0; [..|apply gpaco0_def_mon]. pstep.
eapply gf_mon. apply H0. intros.
apply rclo0_rclo. eapply rclo0_mon. apply PR.
intros. destruct PR0.
- apply rclo0_base. right. apply CIH. apply H.
- destruct H; [destruct H|].
+ apply rclo0_base. right. apply CIH0. left. apply H.
+ apply IN in H. destruct H.
eapply rclo0_mon. apply IN0.
intros. destruct PR0.
* right. apply CIH. apply H.
* right. apply CIH0. right. apply H.
+ apply rclo0_base. right. apply CIH0. right. apply H.
Qed.
|
Lemma
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
gpaco0_cofix
| |
gpaco0_gupaco clo r rg:
gupaco0 clo (gpaco0 clo r rg) <0= gpaco0 clo r rg.
Proof.
eapply gpaco0_cofix.
intros. destruct PR. econstructor.
apply rclo0_rclo. eapply rclo0_mon. apply IN.
intros. destruct PR.
- apply rclo0_base. left.
eapply paco0_mon. apply H.
intros. left; apply CIH.
econstructor. apply rclo0_base. right.
destruct PR; apply H0.
- destruct H. eapply rclo0_mon. apply IN0.
intros. destruct PR; [| right; apply H].
left. eapply paco0_mon. apply H.
intros. destruct PR.
+ left. apply INC. apply H0.
+ right. apply H0.
Qed.
|
Lemma
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
gpaco0_gupaco
| |
gpaco0_gpaco clo r rg:
gpaco0 clo (gpaco0 clo r rg) (gupaco0 clo (rg \0/ r)) <0= gpaco0 clo r rg.
Proof.
intros. apply gpaco0_unfold in PR.
econstructor. apply rclo0_rclo. eapply rclo0_mon. apply PR. clear PR. intros.
destruct PR; [|destruct H; apply IN].
apply rclo0_base. left. pstep.
eapply gf_mon. apply H. clear H. intros.
cut (@gupaco0 clo (rg \0/ r)).
{ intros. destruct H. eapply rclo0_mon. apply IN. intros.
destruct PR0; [|right; apply H].
left. eapply paco0_mon. apply H. intros. destruct PR0; apply H0.
}
apply gpaco0_gupaco. eapply gupaco0_mon. apply PR. intros.
destruct PR0; [apply H|].
eapply gpaco0_mon; [apply H|right|left]; intros; apply PR0.
Qed.
|
Lemma
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
gpaco0_gpaco
| |
gpaco0_uclo uclo clo r rg
(LEclo: uclo <1= gupaco0 clo) :
uclo (gpaco0 clo r rg) <0= gpaco0 clo r rg.
Proof.
intros. apply gpaco0_gupaco. apply LEclo, PR.
Qed.
|
Lemma
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
gpaco0_uclo
| |
gpaco0_weaken clo r rg:
gpaco0 (gupaco0 clo) r rg <0= gpaco0 clo r rg.
Proof.
intros. apply gpaco0_unfold in PR.
induction PR.
- destruct IN; cycle 1. apply gpaco0_base, H.
apply gpaco0_step_gen. eapply gf_mon. apply H.
clear H.
eapply gpaco0_cofix. intros.
apply gpaco0_unfold in PR.
induction PR.
+ destruct IN; cycle 1. apply gpaco0_base, H.
apply gpaco0_step. eapply gf_mon. apply H.
intros. apply gpaco0_base. apply CIH.
eapply gupaco0_mon. apply PR.
intros. destruct PR0; apply H0.
+ apply gpaco0_gupaco.
eapply gupaco0_mon. apply IN. apply H.
- apply gpaco0_gupaco.
eapply gupaco0_mon. apply IN. apply H.
Qed.
|
Lemma
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
gpaco0_weaken
| |
GeneralMonotonicity .
Variable gf: rel -> rel.
|
Section
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
GeneralMonotonicity
| |
gpaco0_mon_gen (gf' clo clo': rel -> rel) r r' rg rg'
(IN: @gpaco0 gf clo r rg)
(gf_mon: monotone0 gf)
(LEgf: gf <1= gf')
(LEclo: clo <1= clo')
(LEr: r <0= r')
(LErg: rg <0= rg') :
@gpaco0 gf' clo' r' rg'.
Proof.
eapply gpaco0_mon; [|apply LEr|apply LErg].
destruct IN. econstructor.
eapply rclo0_mon_gen. apply IN. apply LEclo.
intros. destruct PR; [| right; apply H].
left. eapply paco0_mon_gen. apply H.
- intros. eapply LEgf.
eapply gf_mon. apply PR.
intros. eapply rclo0_mon_gen. apply PR0. apply LEclo. intros; apply PR1.
- intros. apply PR.
Qed.
|
Lemma
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
gpaco0_mon_gen
| |
gpaco0_mon_bot (gf' clo clo': rel -> rel) r' rg'
(IN: @gpaco0 gf clo bot0 bot0)
(gf_mon: monotone0 gf)
(LEgf: gf <1= gf')
(LEclo: clo <1= clo'):
@gpaco0 gf' clo' r' rg'.
Proof.
eapply gpaco0_mon_gen. apply IN. apply gf_mon. apply LEgf. apply LEclo. contradiction. contradiction.
Qed.
|
Lemma
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
gpaco0_mon_bot
| |
gupaco0_mon_gen (gf' clo clo': rel -> rel) r r'
(IN: @gupaco0 gf clo r)
(gf_mon: monotone0 gf)
(LEgf: gf <1= gf')
(LEclo: clo <1= clo')
(LEr: r <0= r'):
@gupaco0 gf' clo' r'.
Proof.
eapply gpaco0_mon_gen. apply IN. apply gf_mon. apply LEgf. apply LEclo. apply LEr. apply LEr.
Qed.
|
Lemma
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
gupaco0_mon_gen
| |
Compatibility .
Variable gf: rel -> rel.
Hypothesis gf_mon: monotone0 gf.
Structure compatible0 (clo: rel -> rel) : Prop :=
compat0_intro {
compat0_mon: monotone0 clo;
compat0_compat : forall r,
clo (gf r) <0= gf (clo r);
}.
Structure wcompatible0 clo : Prop :=
wcompat0_intro {
wcompat0_mon: monotone0 clo;
wcompat0_wcompat : forall r,
clo (gf r) <0= gf (gupaco0 gf clo r);
}.
|
Section
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
Compatibility
| |
rclo0_dist clo
(MON: monotone0 clo)
(DIST: forall r1 r2, clo (r1 \0/ r2) <0= (clo r1 \0/ clo r2)):
forall r1 r2, rclo0 clo (r1 \0/ r2) <0= (rclo0 clo r1 \0/ rclo0 clo r2).
Proof.
intros. induction PR.
+ destruct IN; [left|right]; apply rclo0_base, H.
+ assert (REL: clo (rclo0 clo r1 \0/ rclo0 clo r2)).
{ eapply MON. apply IN. apply H. }
apply DIST in REL. destruct REL; [left|right]; apply rclo0_clo, H0.
Qed.
|
Lemma
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
rclo0_dist
| |
rclo0_compat clo
(COM: compatible0 clo):
compatible0 (rclo0 clo).
Proof.
econstructor.
- apply rclo0_mon.
- intros. induction PR.
+ eapply gf_mon. apply IN.
intros. eapply rclo0_base. apply PR.
+ eapply gf_mon.
* eapply COM. eapply COM. apply IN. apply H.
* intros. eapply rclo0_clo. apply PR.
Qed.
|
Lemma
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
rclo0_compat
| |
rclo0_wcompat clo
(COM: wcompatible0 clo):
wcompatible0 (rclo0 clo).
Proof.
econstructor.
- apply rclo0_mon.
- intros. induction PR.
+ eapply gf_mon. apply IN.
intros. apply gpaco0_base. apply PR.
+ eapply gf_mon.
* eapply COM. eapply COM. apply IN. apply H.
* intros. eapply gpaco0_gupaco. apply gf_mon.
eapply gupaco0_mon_gen; intros; [apply PR|apply gf_mon|apply PR0| |apply PR0].
apply rclo0_clo_base, PR0.
Qed.
|
Lemma
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
rclo0_wcompat
| |
compat0_wcompat clo
(CMP: compatible0 clo):
wcompatible0 clo.
Proof.
econstructor. apply CMP.
intros. apply CMP in PR.
eapply gf_mon. apply PR.
intros. apply gpaco0_clo, PR0.
Qed.
|
Lemma
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
compat0_wcompat
| |
wcompat0_compat clo
(WCMP: wcompatible0 clo) :
compatible0 (gupaco0 gf clo).
Proof.
econstructor.
{ red; intros. eapply gpaco0_mon. apply IN. apply LE. apply LE. }
intros. apply gpaco0_unfold in PR; [|apply gf_mon].
induction PR.
- destruct IN; cycle 1.
+ eapply gf_mon. apply H.
intros. apply gpaco0_base, PR.
+ eapply gf_mon. apply H.
intros. apply gpaco0_gupaco. apply gf_mon.
eapply gupaco0_mon. apply PR.
intros. apply gpaco0_step. apply gf_mon.
eapply gf_mon. destruct PR0 as [X|X]; apply X.
intros. apply gpaco0_base, PR1.
- eapply gf_mon, gpaco0_gupaco, gf_mon.
apply WCMP. eapply WCMP. apply IN.
intros. apply H, PR.
Qed.
|
Lemma
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
wcompat0_compat
| |
wcompat0_union clo1 clo2
(WCMP1: wcompatible0 clo1)
(WCMP2: wcompatible0 clo2):
wcompatible0 (clo1 \1/ clo2).
Proof.
econstructor.
- apply monotone0_union. apply WCMP1. apply WCMP2.
- intros. destruct PR.
+ apply WCMP1 in H. eapply gf_mon. apply H.
intros. eapply gupaco0_mon_gen. apply PR. apply gf_mon.
intros; apply PR0. left; apply PR0. intros; apply PR0.
+ apply WCMP2 in H. eapply gf_mon. apply H.
intros. eapply gupaco0_mon_gen. apply PR. apply gf_mon.
intros; apply PR0. right; apply PR0. intros; apply PR0.
Qed.
|
Lemma
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
wcompat0_union
| |
Soundness .
Variable gf: rel -> rel.
Hypothesis gf_mon: monotone0 gf.
|
Section
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
Soundness
| |
gpaco0_compat_init clo
(CMP: compatible0 gf clo):
gpaco0 gf clo bot0 bot0 <0= paco0 gf bot0.
Proof.
intros. destruct PR. revert IN.
pcofix CIH. intros.
pstep. eapply gf_mon; [| right; apply CIH, rclo0_rclo, PR].
apply compat0_compat with (gf:=gf). apply rclo0_compat. apply gf_mon. apply CMP.
eapply rclo0_mon. apply IN.
intros. destruct PR; [|contradiction]. _punfold H; [..|apply gpaco0_def_mon, gf_mon].
eapply gpaco0_def_mon. apply gf_mon. apply H.
intros. destruct PR; [|destruct H0; contradiction]. left. apply H0.
Qed.
|
Lemma
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
gpaco0_compat_init
| |
gpaco0_init clo
(WCMP: wcompatible0 gf clo):
gpaco0 gf clo bot0 bot0 <0= paco0 gf bot0.
Proof.
intros. eapply gpaco0_compat_init.
- apply wcompat0_compat, WCMP. apply gf_mon.
- eapply gpaco0_mon_bot. apply PR. apply gf_mon. intros; apply PR0.
intros. apply gpaco0_clo, PR0.
Qed.
|
Lemma
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
gpaco0_init
| |
gpaco0_unfold_bot clo
(WCMP: wcompatible0 gf clo):
gpaco0 gf clo bot0 bot0 <0= gf (gpaco0 gf clo bot0 bot0).
Proof.
intros. apply gpaco0_init in PR; [|apply WCMP].
_punfold PR; [..|apply gf_mon].
eapply gf_mon. apply PR.
intros. destruct PR0; [|contradiction]. apply gpaco0_final. apply gf_mon. right. apply H.
Qed.
|
Lemma
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
gpaco0_unfold_bot
| |
Distributivity .
Variable gf: rel -> rel.
Hypothesis gf_mon: monotone0 gf.
|
Section
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
Distributivity
| |
gpaco0_dist clo r rg
(CMP: wcompatible0 gf clo)
(DIST: forall r1 r2, clo (r1 \0/ r2) <0= (clo r1 \0/ clo r2)):
gpaco0 gf clo r rg <0= (paco0 gf (rclo0 clo (rg \0/ r)) \0/ rclo0 clo r).
Proof.
intros. apply gpaco0_unfold in PR; [|apply gf_mon].
apply rclo0_dist in PR; [|apply CMP|apply DIST].
destruct PR; [|right; apply H].
left. revert H.
pcofix CIH; intros.
apply rclo0_wcompat in H0; [|apply gf_mon|apply CMP].
pstep. eapply gf_mon. apply H0. clear H0. intros.
apply gpaco0_unfold in PR; [|apply gf_mon].
apply rclo0_compose in PR.
apply rclo0_dist in PR; [|apply CMP|apply DIST].
destruct PR.
- right. apply CIH.
eapply rclo0_mon. apply H. intros.
eapply gf_mon. apply PR. intros.
apply gpaco0_gupaco. apply gf_mon.
apply gpaco0_gen_rclo. apply gf_mon.
eapply gupaco0_mon. apply PR0. intros.
destruct PR1; apply H0.
- assert (REL: @rclo0 clo (rclo0 clo (gf (gupaco0 gf clo ((rg \0/ r) \0/ (rg \0/ r))) \0/ (rg \0/ r)))).
{ eapply rclo0_mon. apply H. intros. apply gpaco0_unfold in PR. apply PR. apply gf_mon. }
apply rclo0_rclo in REL.
apply rclo0_dist in REL; [|apply CMP|apply DIST].
right. destruct REL; cycle 1.
+ apply CIH0, H0.
+ apply CIH.
eapply rclo0_mon. apply H0. intros.
eapply gf_mon. apply PR. intros.
eapply gupaco0_mon. apply PR0. intros.
destruct PR1; apply H1.
Qed.
|
Lemma
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
gpaco0_dist
| |
gpaco0_dist_reverse clo r rg:
(paco0 gf (rclo0 clo (rg \0/ r)) \0/ rclo0 clo r) <0= gpaco0 gf clo r rg.
Proof.
intros. destruct PR; cycle 1.
- eapply gpaco0_rclo. apply H.
- econstructor. apply rclo0_base. left.
revert H. pcofix CIH; intros.
_punfold H0; [|apply gf_mon]. pstep.
eapply gf_mon. apply H0. intros.
destruct PR.
+ apply rclo0_base. right. apply CIH, H.
+ eapply rclo0_mon. apply H. intros.
right. apply CIH0. apply PR.
Qed.
|
Lemma
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
gpaco0_dist_reverse
| |
Companion .
Variable gf: rel -> rel.
Hypothesis gf_mon: monotone0 gf.
|
Section
|
root
|
[
"Require Export Program.",
"From Paco Require Import paco0 pacotac."
] |
gpaco0.v
|
Companion
|
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