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Coq-Paco

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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
End of preview. Expand in Data Studio

Coq-Paco

Structured declarations from the Paco library - a Coq library for parameterized coinduction. Source: github.com/snu-sf/paco

Schema

Column Type Description
fact string Declaration body (without type keyword)
type string Declaration type (Lemma, Definition, etc.)
library string Source module
imports list Import statements
filename string Source file path
symbolic_name string Declaration identifier

Statistics

  • Total entries: 2183
  • Unique files: 39
  • Declaration types: Class, CoInductive, Corollary, Definition, Global Instance, Inductive, Lemma, Let, Local Notation, Ltac, Record, Section, Tactic, Theorem, Variant

Usage 

from datasets import load_dataset
ds = load_dataset("phanerozoic/Coq-Paco")

License

bsd-3-clause

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Number of rows:
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Collection including phanerozoic/Coq-Paco