fact stringlengths 14 11.5k | type stringclasses 22
values | library stringclasses 7
values | imports listlengths 0 7 | filename stringclasses 136
values | symbolic_name stringlengths 1 36 | docstring stringclasses 1
value |
|---|---|---|---|---|---|---|
bind_iforest_stronger {E} : forall A B, iforest E A -> (A -> iforest E B) -> iforest E B := fun A B (PA: iforest E A) (K: A -> iforest E B) (tb: itree E B) => exists (ta: itree E A) (k: A -> itree E B), PA ta /\ tb ≈ bind ta k /\ (forall a, Leaf a ta -> K a (k a)). (** Alternate, logically equivalent version of [bind_i... | Definition | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | bind_iforest_stronger | |
bind_iforest' {E} : forall A B, iforest E A -> (A -> iforest E B) -> iforest E B := fun A B (PA: iforest E A) (K: A -> iforest E B) (tb: itree E B) => exists (ta: itree E A), PA ta /\ ((exists (k: A -> itree E B), (forall a, Leaf a ta -> K a (k a)) /\ tb ≈ bind ta k) \/ (forall k, (forall a, K a (k a)) /\ tb ≈ bind ta ... | Definition | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | bind_iforest' | |
bind_iforest_bind_iforest' {E}: forall A B PA K (tb : itree E B), bind_iforest A B PA K tb <-> bind_iforest' A B PA K tb. Proof. intros. split. intros. - red. red in H. destruct H as (ta & ka & HPA & eq & HR). exists ta. split; auto. left. exists ka. split; auto. - intros. red. red in H. destruct H as (ta & EQ1 & [(k &... | Lemma | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | bind_iforest_bind_iforest' | |
handler_correct {E F} (h_spec: E ~> iforest F) (h: E ~> itree F) : Prop := (forall T e, h_spec T e (h T e)). | Definition | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | handler_correct | |
interp_iforestF {E F} (h_spec : forall T, E T -> itree F T -> Prop) {R : Type} (RR : relation R) (sim : itree E R -> itree F R -> Prop) : itree' E R -> itree F R -> Prop := | Interp_iforest_Ret : forall r1 r2 (REL: RR r1 r2) (t2 : itree F R) (eq2 : t2 ≈ (Ret r2)), interp_iforestF h_spec RR sim (RetF r1) t2 | Interp_ifo... | Inductive | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | interp_iforestF | |
interp_iforestF_mono E F h_spec R RR (t0 : itree' E R) (t1 : itree F R) sim sim' (IN : interp_iforestF h_spec RR sim t0 t1) (LE : sim <2= sim') : (interp_iforestF h_spec RR sim' t0 t1). Proof. induction IN; eauto with itree. Qed. #[global] Hint Resolve interp_iforestF_mono : paco. | Lemma | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | interp_iforestF_mono | |
interp_iforest_ E F h_spec R RR sim (t0 : itree E R) (t1 : itree F R) : Prop := interp_iforestF h_spec RR sim (observe t0) t1. #[global] Hint Unfold interp_iforest_ : itree. | Definition | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | interp_iforest_ | |
interp_iforest__mono E F h_spec R RR : monotone2 (interp_iforest_ E F h_spec R RR). Proof. do 2 red. intros. eapply interp_iforestF_mono; eauto. Qed. #[global] Hint Resolve interp_iforest__mono : paco. (* Definition 5.2 *) | Lemma | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | interp_iforest__mono | |
interp_iforest {E F} (h_spec : E ~> iforest F) : forall R (RR: relation R), itree E R -> iforest F R := fun R (RR: relation R) => paco2 (interp_iforest_ E F h_spec R RR) bot2. (* Figure 7: Interpreter law for Ret *) | Definition | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | interp_iforest | |
interp_iforest_ret : forall R E F (h_spec : E ~> iforest F) (r : R) , Eq1_iforest _ (interp_iforest h_spec R eq (ret r)) (ret r). Proof. intros. repeat red. split; [| split]. - intros. split; intros. + unfold interp_iforest in H0. pinversion H0. subst. cbn. rewrite <- H. assumption. + pstep. econstructor. reflexivity. ... | Lemma | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | interp_iforest_ret | |
inj_pair2 : forall (U : Type) (P : U -> Type) (p : U) (x y : P p), existT P p x = existT P p y -> x = y. Proof. intros. apply JMeq.JMeq_eq. refine ( match H in _ = w return JMeq.JMeq x (projT2 w) with | eq_refl => JMeq.JMeq_refl end). Qed. #[global] Instance interp_iforest_Proper3 {E F} (h_spec : E ~> iforest F) R RR :... | Lemma | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | inj_pair2 | |
interp_iforest_correct_exec : forall {E F} (h_spec: E ~> iforest F) (h: E ~> itree F), handler_correct h_spec h -> forall R RR `{Reflexive _ RR} t t', t ≈ t' -> interp_iforest h_spec R RR t (interp h t'). Proof. intros. revert t t' H1. pcofix CIH. intros t t' eq. pstep. red. unfold interp, Basics.iter, MonadIter_itree.... | Lemma | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | interp_iforest_correct_exec | |
Leaf_Vis_sub : forall {E} {R} X (e : E X) (k : X -> itree E R) u x, Leaf u (k x) -> Leaf u (Vis e k). Proof. intros. eapply LeafVis. reflexivity. apply H. Qed. | Lemma | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | Leaf_Vis_sub | |
eutt_Leaf_ : forall {E} {R} (RR : R -> Prop) (ta : itree E R) (IN: forall (a : R), Leaf a ta -> RR a), eutt (fun u1 u2 => u1 = u2 /\ RR u1) ta ta. Proof. intros E R. ginit. gcofix CIH; intros. setoid_rewrite (itree_eta ta) in IN. gstep. red. destruct (observe ta). - econstructor. split; auto. apply IN. econstructor. re... | Lemma | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | eutt_Leaf_ | |
eutt_Leaf : forall E R (ta : itree E R), eutt (fun u1 u2 => u1 = u2 /\ Leaf u1 ta) ta ta. Proof. intros. apply eutt_Leaf_. auto. Qed. (* Figure 7: interp Trigger law *) (* morally, we should only work with "proper" triggers everywhere *) | Lemma | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | eutt_Leaf | |
interp_iforest_trigger : forall E F (h_spec : E ~> iforest F) R (e : E R) (HP : forall T, Proper (eq ==> Eq1_iforest T) (h_spec T)) , Eq1_iforest _ (interp_iforest h_spec R eq (trigger e)) (h_spec R e). Proof. intros. red. split; [| split]. - intros; split; intros. + unfold trigger in H0. red in H0. pinversion H0; subs... | Lemma | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | interp_iforest_trigger | |
interp_iforest_spin_accepts_anything : forall E F (h_spec : E ~> iforest F) R RR (t : itree F R), interp_iforest h_spec R RR ITree.spin t. Proof. intros. pcofix CIH. pstep. red. cbn. econstructor. right. apply CIH. Qed. (* Figure 7: Structural law for tau *) | Lemma | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | interp_iforest_spin_accepts_anything | |
interp_iforest_tau : forall E F (h_spec : E ~> iforest F) R RR (t_spec : itree E R), Eq1_iforest _ (interp_iforest h_spec R RR t_spec) (interp_iforest h_spec R RR (Tau t_spec)). Proof. intros. split; [| split]. - intros; split; intros. + rewrite <- H. pstep. red. econstructor. left. apply H0. + rewrite H. pinversion H0... | Lemma | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | interp_iforest_tau | |
interp_iforest_ret_inv : forall E F (h_spec : E ~> iforest F) R RR (r1 : R) (t : itree F R) (H : interp_iforest h_spec R RR (ret r1) t), exists r2, RR r1 r2 /\ t ≈ ret r2. Proof. intros. punfold H. red in H. inversion H; subst. exists r2; eauto. Qed. | Lemma | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | interp_iforest_ret_inv | |
interp_iforest_vis_inv : forall E F (h_spec : E ~> iforest F) R RR S (e : E S) (k : S -> itree E R) (t : itree F R) (H : interp_iforest h_spec R RR (vis e k) t), exists ms, exists (ks : S -> itree F R), h_spec S e ms /\ t ≈ (bind ms ks). Proof. intros. punfold H. red in H. inversion H; subst. apply inj_pair2 in H2. app... | Lemma | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | interp_iforest_vis_inv | |
interp_iforest_tau_inv : forall E F (h_spec : E ~> iforest F) R RR (s : itree E R) (t : itree F R) (H : interp_iforest h_spec R RR (Tau s) t), interp_iforest h_spec R RR s t. Proof. intros. punfold H. red in H. inversion H; subst. pclearbot. apply HS. Qed. | Lemma | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | interp_iforest_tau_inv | |
case_iforest_handler_correct : forall {E1 E2 F} (h1_spec: E1 ~> iforest F) (h2_spec: E2 ~> iforest F) (h1: E1 ~> itree F) (h2: E2 ~> itree F) (C1: handler_correct h1_spec h1) (C2: handler_correct h2_spec h2), handler_correct (case_ h1_spec h2_spec) (case_ h1 h2). Proof. intros E1 E2 F h1_spec h2_spec h1 h2 C1 C2. unfol... | Lemma | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | case_iforest_handler_correct | |
iforest_compose {F G : Type -> Type} {T : Type} (TT : relation T) (g_spec : F ~> iforest G) (PF: iforest F T) : iforest G T := fun (g:itree G T) => exists f : itree F T, PF f /\ (interp_iforest g_spec) T TT f g. | Definition | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | iforest_compose | |
handler_correct_iforest {E F G} (h_spec: E ~> iforest F) (h: E ~> itree F) (g_spec: F ~> iforest G) (g: F ~> itree G) := (forall T TT e, (iforest_compose TT g_spec (h_spec T e)) (interp g (h T e))). | Definition | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | handler_correct_iforest | |
singletonT {E}: itree E ~> iforest E := fun R t t' => t' ≈ t. | Definition | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | singletonT | |
iter_cont {I E R} (step' : I -> itree E (I + R)) : I + R -> itree E R := fun lr => ITree.on_left lr l (Tau (ITree.iter step' l)). #[global] Polymorphic Instance MonadIter_iforest {E} : MonadIter (iforest E) := fun R I (step : I -> iforest E (I + R)) i => fun (r : itree E R) => (exists (step' : I -> (itree E (I + R)%typ... | Definition | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | iter_cont | |
eqit_Leaf_bind_clo b1 b2 (r : itree E R -> itree E S -> Prop) : itree E R -> itree E S -> Prop := | pbc_intro_h U (t1 t2: itree E U) (k1 : U -> itree E R) (k2 : U -> itree E S) (EQV: eqit eq b1 b2 t1 t2) (REL: forall u, Leaf u t1 -> r (k1 u) (k2 u)) : eqit_Leaf_bind_clo b1 b2 r (ITree.bind t1 k1) (ITree.bind t2 k2) . H... | Inductive | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | eqit_Leaf_bind_clo | |
eqit_Leaf_clo_bind (RS : R -> S -> Prop) b1 b2 vclo (MON: monotone2 vclo) (CMP: compose (eqitC RS b1 b2) vclo <3= compose vclo (eqitC RS b1 b2)) (ID: id <3= vclo): eqit_Leaf_bind_clo b1 b2 <3= gupaco2 (eqit_ RS b1 b2 vclo) (eqitC RS b1 b2). Proof. gcofix CIH. intros. destruct PR. guclo eqit_clo_trans. econstructor; aut... | Lemma | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | eqit_Leaf_clo_bind | |
eqit_Leaf_bind' {E} {R} {T} b1 b2 (t1 t2: itree E T) (k1 k2: T -> itree E R) : eqit eq b1 b2 t1 t2 -> (forall r, Leaf r t1 -> eqit eq b1 b2 (k1 r) (k2 r)) -> @eqit E _ _ eq b1 b2 (ITree.bind t1 k1) (ITree.bind t2 k2). Proof. intros. ginit. guclo (@eqit_Leaf_clo_bind E R R eq). unfold eqit in *. econstructor; eauto with... | Lemma | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | eqit_Leaf_bind' | |
eqit_Leaf_bind'' {E} {R S} {T} (RS : R -> S -> Prop) b1 b2 (t1 t2: itree E T) (k1: T -> itree E R) (k2 : T -> itree E S) : eqit eq b1 b2 t1 t2 -> (forall r, Leaf r t1 -> eqit RS b1 b2 (k1 r) (k2 r)) -> @eqit E _ _ RS b1 b2 (ITree.bind t1 k1) (ITree.bind t2 k2). Proof. intros. ginit. guclo (@eqit_Leaf_clo_bind E R S RS)... | Lemma | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | eqit_Leaf_bind'' | |
eutt_ret_vis_abs : forall {X Y E} (x: X) (e: E Y) k, Ret x ≈ Vis e k -> False. Proof. intros. punfold H; inv H. Qed. | Lemma | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | eutt_ret_vis_abs | |
simpl_iter := unfold iter, Iter_Kleisli, Basics.iter, MonadIter_itree. | Ltac | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | simpl_iter | |
g {a b : Type} {E} (x0 : a * nat -> itree E (a + b)) (a1 : a) := (fun '(x, k) => bind (x0 (x, k)) (fun ir : a + b => match ir with | inl i0 => ret (inl (i0, k)) | inr r0 => ret (inr r0) end)). | Definition | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | g | |
f {a b : Type} {E} : a * nat -> itree E (a * nat + b) := fun '(x, k) => ret (inl ((x, S k))). | Definition | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | f | |
iter_succ_dinatural : forall {a b : Type} {E} (x0 : a * nat -> itree E (a + b)) (a1 : a), iter (C := Kleisli (itree E)) (bif := sum) (f >>> case_ (g x0 a1) inr_) ⩯ f >>> case_ (iter (C := Kleisli (itree E)) (bif := sum) ((g x0 a1) >>> (case_ f inr_))) (id_ _). Proof. intros. rewrite iter_dinatural. reflexivity. Qed. | Lemma | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | iter_succ_dinatural | |
iter_eq_start_index : forall (a b : Type) (E : Type -> Type) (x0 : a * nat -> itree E (a + b)) (a1 : a), iter (C := Kleisli (itree E)) (bif := sum) (fun '(x, k) => bind (x0 (x, S k)) (fun ir : a + b => match ir with | inl i0 => ret (inl (i0, S k)) | inr r0 => ret (inr r0) end)) (a1, 0) ≈ iter (C := Kleisli (itree E)) (... | Lemma | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | iter_eq_start_index | |
Eq1_iforest' {E} : Eq1 (iforest E) := fun a PA PA' => (forall x, (PA x -> exists y, x ≈ y /\ PA' y)) /\ (forall y, (PA' y -> exists x, x ≈ y /\ PA x)) /\ eutt_closed PA /\ eutt_closed PA'. | Definition | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | Eq1_iforest' | |
Eq1_iforest'_Eq1_iforest : forall E a PA PA', @Eq1_iforest E a PA PA' -> Eq1_iforest' a PA PA'. Proof. intros. red. red in H. destruct H as (HXY & EPA & EPA'). split. intros. exists x. split; [reflexivity|]. specialize (HXY x x). apply HXY. reflexivity. assumption. split; try tauto. intros. exists y. split; [reflexivit... | Lemma | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | Eq1_iforest'_Eq1_iforest | |
ret_bind : forall {E} (a b : Type) (f : a -> iforest E b) (x : a), eutt_closed (f x) -> eq1 (bind (ret x) f) (f x). Proof. intros. split; [| split]. - intros t t' eq; split; intros eqtt'. * cbn in *. repeat red in eqtt'. destruct eqtt' as (ta & k & EQ1 & EQ2 & KA). + unfold bind, Monad_itree in EQ2. rewrite EQ1, Eqit.b... | Lemma | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | ret_bind | |
agrees_itree := (eutt (fun a p => p a)) (only parsing). | Notation | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | agrees_itree | |
bind_stronger {E A B} (PA: iforest E A) (K: A -> iforest E B) : iforest E B := fun (tb: itree E B) => exists (ta: itree E A), PA ta /\ exists (k: A -> itree E B), (agrees_itree (fmap k ta) (fmap K ta) /\ tb ≈ bind ta k). | Definition | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | bind_stronger | |
agree_itree_Leaf E A B (ta : itree E A) (K : A -> iforest E B) (k : A -> itree E B) : (forall a, Leaf a ta -> K a (k a)) <-> (agrees_itree (fmap k ta) (fmap K ta)). Proof. split; intros. - cbn. red. unfold ITree.map. eapply eqit_Leaf_bind''. + reflexivity. + intros. apply eqit_Ret. apply H. assumption. - revert H. indu... | Lemma | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | agree_itree_Leaf | |
distinguish_bind {E} {A B} (a : A) (ma : itree E A) (k1 k2 : A -> itree E B) (HRET : Leaf a ma) (NEQ: ~((k1 a) ≈ (k2 a))) : not ((ITree.bind ma k1) ≈ (ITree.bind ma k2)). Proof. intros HI; eapply eqit_bind_Leaf_inv in HI; eauto. Qed. | Lemma | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | distinguish_bind | |
not_Leaf {E} {A B} : inhabited B -> forall (ta: itree E A), (exists tb, forall (k : A -> itree E B), tb ≈ bind ta k) -> forall (a:A), ~ Leaf a ta. Proof. intros [b] ta [tb HK] a HRet. revert tb HK; induction HRet; intros tb HK. - setoid_rewrite unfold_bind in HK. setoid_rewrite H in HK. generalize (HK (fun _ => ITree.s... | Lemma | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | not_Leaf | |
bind_ret : forall {E} (A : Type) (PA : iforest E A), eutt_closed PA -> eq1 (bind PA (fun x => ret x)) PA. Proof. intros. split; [| split]. + intros t t' eq; split; intros eqtt'. * cbn in *. destruct eqtt' as (ta & k & HPA & EQ & HRET). eapply H; [symmetry; eauto | clear eq t']. eapply H; [eauto | clear EQ t]. eapply H;... | Lemma | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | bind_ret | |
EQ_REL {E A} (ta : itree E A) : A -> A -> Prop := fun a b => a = b /\ Leaf a ta. | Definition | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | EQ_REL | |
Symmteric_EQ_REL {E A} (ta : itree E A) : Symmetric (EQ_REL ta). Proof. repeat red. intros a b (EQ & H). split. - symmetry. assumption. - subst; auto. Qed. | Lemma | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | Symmteric_EQ_REL | |
Transitive_EQ_REL {E A} (ta : itree E A) : Transitive (EQ_REL ta). Proof. repeat red. intros a b c (EQ1 & H1) (EQ2 & H2). split. - rewrite EQ1. assumption. - assumption. Qed. #[global] Instance EQ_REL_Proper {E A} : Proper (eutt eq ==> eq ==> eq ==> iff) (@EQ_REL E A). Proof. repeat red. intros. subst. split; intros; u... | Lemma | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | Transitive_EQ_REL | |
eq_relation {A} (R S : A -> A -> Prop) := R <2= S /\ S <2= R. #[global] Instance eutt_EQ_REL_Proper {E} {A} : Proper (eq_relation ==> eutt eq ==> @eutt E A A eq ==> iff) (eutt). Proof. repeat red. intros; split; intros. - rewrite <- H0. rewrite <- H1. clear H0 H1. destruct H. eapply eqit_mon; eauto. - rewrite H0, H1. d... | Definition | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | eq_relation | |
eutt_EQ_REL_Reflexive_ {E} {A} (ta : itree E A) : forall R, (EQ_REL ta) <2= R -> eutt R ta ta. Proof. revert ta. ginit. gcofix CIH. intros ta HEQ. gstep. red. genobs ta obs. destruct obs. - econstructor. apply HEQ. red. split; auto. rewrite itree_eta. rewrite <- Heqobs. constructor 1. reflexivity. - econstructor. gbase... | Lemma | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | eutt_EQ_REL_Reflexive_ | |
eutt_EQ_REL_Reflexive {E} {A} (ta : itree E A) : eutt (EQ_REL ta) ta ta. Proof. apply eutt_EQ_REL_Reflexive_. auto. Qed. | Lemma | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | eutt_EQ_REL_Reflexive | |
RET_EQ {E} {A} (ta : itree E A) : A -> A -> Prop := fun x y => Leaf x ta /\ Leaf y ta. (* Figure 7: 3rd monad law, one direction bind associativity *) | Definition | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | RET_EQ | |
bind_bind_iforest : forall {E} (A B C : Type) (PA : iforest E A) (KB : A -> iforest E B) (KC : B -> iforest E C) (PQOK : eutt_closed PA) (KBP : Proper (eq ==> eq1) KB) (KCP : Proper (eq ==> eq1) KC) (t : itree E C), (bind (bind PA KB) KC) t -> (bind PA (fun a => bind (KB a) KC)) t. Proof. (* PA ~a> KB a ~b> KC b *) int... | Lemma | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | bind_bind_iforest | |
ND : Type -> Prop := | Pick : ND bool. | Inductive | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | ND | |
PA : iforest ND bool := fun (ta : itree ND bool) => ta ≈ trigger Pick. | Definition | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | PA | |
KB : bool -> iforest ND bool := fun b => PA. | Definition | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | KB | |
KC : bool -> iforest ND bool := fun b => if b then (fun tc : itree ND bool => (tc ≈ ret true) \/ (tc ≈ ret false)) else (fun tc : itree ND bool => tc ≈ ITree.spin). | Definition | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | KC | |
t : itree ND bool := bind (trigger Pick) (fun (b:bool) => if b then bind (trigger Pick) (fun (x:bool) => if x then ret true else ITree.spin) else bind (trigger Pick) (fun (x:bool) => if x then ret false else ITree.spin)). | Definition | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | t | |
bind_right_assoc : bind PA (fun a => bind (KB a) KC) t. Proof. repeat red. exists (trigger Pick). exists (fun (b:bool) => if b then (bind (trigger Pick) (fun (x:bool) => if x then ret true else ITree.spin)) else (bind (trigger Pick) (fun (x:bool) => if x then ret false else ITree.spin))). split; auto. red. reflexivity.... | Lemma | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | bind_right_assoc | |
not_bind_left_assoc : ~ (bind (bind PA KB) KC t). Proof. intro H. repeat red in H. destruct H as (ta & k & HB & HEQ & HRET). destruct HB as (tb & kb & HX & HEQ' & HRET'). red in HX. rewrite HX in *. setoid_rewrite HX in HRET'. clear tb HX. rewrite HEQ' in HEQ. unfold t in HEQ. unfold bind, Monad_itree in HEQ. rewrite E... | Lemma | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | not_bind_left_assoc | |
bind_bind_counterexample : exists t, bind PA (fun a => bind (KB a) KC) t /\ ~ (bind (bind PA KB) KC t). Proof. exists t. split. apply bind_right_assoc. apply not_bind_left_assoc. Qed. | Lemma | extra | [
"From Paco Require Import paco.",
"From ExtLib Require Import\n Structures."
] | extra/IForest.v | bind_bind_counterexample | |
aexp : Type := | ANum (n : nat) | AId (x : string) | APlus (a1 a2 : aexp) | AMinus (a1 a2 : aexp) | AMult (a1 a2 : aexp). | Inductive | hoare_example | [
"From Coq Require Import\n Arith.",
"From ExtLib Require Import\n Data.",
"From ITree Require Import\n Events.",
"From ExtLib Require Import\n Core."
] | hoare_example/Imp.v | aexp | |
bexp : Type := | BTrue | BFalse | BEq (a1 a2 : aexp) | BLe (a1 a2 : aexp) | BNot (b : bexp) | BAnd (b1 b2 : bexp). | Inductive | hoare_example | [
"From Coq Require Import\n Arith.",
"From ExtLib Require Import\n Data.",
"From ITree Require Import\n Events.",
"From ExtLib Require Import\n Core."
] | hoare_example/Imp.v | bexp | |
com : Set := CSkip : com | CAss : string -> aexp -> com | CSeq : com -> com -> com | CIf : bexp -> com -> com -> com | CWhile : bexp -> com -> com. (* ========================================================================== *) (** ** Notations *) | Inductive | hoare_example | [
"From Coq Require Import\n Arith.",
"From ExtLib Require Import\n Data.",
"From ITree Require Import\n Events.",
"From ExtLib Require Import\n Core."
] | hoare_example/Imp.v | com | |
AId : string >-> aexp. | Coercion | hoare_example | [
"From Coq Require Import\n Arith.",
"From ExtLib Require Import\n Data.",
"From ITree Require Import\n Events.",
"From ExtLib Require Import\n Core."
] | hoare_example/Imp.v | AId | |
ANum : nat >-> aexp. | Coercion | hoare_example | [
"From Coq Require Import\n Arith.",
"From ExtLib Require Import\n Data.",
"From ITree Require Import\n Events.",
"From ExtLib Require Import\n Core."
] | hoare_example/Imp.v | ANum | |
bool_to_bexp (b : bool) : bexp := if b then BTrue else BFalse. | Definition | hoare_example | [
"From Coq Require Import\n Arith.",
"From ExtLib Require Import\n Data.",
"From ITree Require Import\n Events.",
"From ExtLib Require Import\n Core."
] | hoare_example/Imp.v | bool_to_bexp | |
bool_to_bexp : bool >-> bexp. Bind Scope imp_scope with aexp. Bind Scope imp_scope with bexp. Delimit Scope imp_scope with imp. | Coercion | hoare_example | [
"From Coq Require Import\n Arith.",
"From ExtLib Require Import\n Data.",
"From ITree Require Import\n Events.",
"From ExtLib Require Import\n Core."
] | hoare_example/Imp.v | bool_to_bexp | |
var := string. (* Definition var : Set := string. *) (** For simplicity, the language manipulates [nat]s as values. *) (* Definition value : Type := nat. *) | Notation | hoare_example | [
"From Coq Require Import\n Arith.",
"From ExtLib Require Import\n Data.",
"From ITree Require Import\n Events.",
"From ExtLib Require Import\n Core."
] | hoare_example/Imp.v | var | |
value := nat. (* (** Expressions are made of variables, constant literals, and arithmetic operations. *) | Notation | hoare_example | [
"From Coq Require Import\n Arith.",
"From ExtLib Require Import\n Data.",
"From ITree Require Import\n Events.",
"From ExtLib Require Import\n Core."
] | hoare_example/Imp.v | value | |
expr : Type := | Var (_ : var) | Lit (_ : value) | Plus (_ _ : expr) | Minus (_ _ : expr) | Mult (_ _ : expr). (** The statements are straightforward. The [While] statement is the only potentially diverging one. *) | Inductive | hoare_example | [
"From Coq Require Import\n Arith.",
"From ExtLib Require Import\n Data.",
"From ITree Require Import\n Events.",
"From ExtLib Require Import\n Core."
] | hoare_example/Imp.v | expr | |
stmt : Type := | Assign (x : var) (e : expr) (* x = e *) | Seq (a b : stmt) (* a ; b *) | If (i : expr) (t e : stmt) (* if (i) then { t } else { e } *) | While (t : expr) (b : stmt) (* while (t) { b } *) | Skip (* ; *) . (* ========================================================================== *) (** ** Notations *... | Inductive | hoare_example | [
"From Coq Require Import\n Arith.",
"From ExtLib Require Import\n Data.",
"From ITree Require Import\n Events.",
"From ExtLib Require Import\n Core."
] | hoare_example/Imp.v | stmt | |
Var_coerce : string -> expr := Var. | Definition | hoare_example | [
"From Coq Require Import\n Arith.",
"From ExtLib Require Import\n Data.",
"From ITree Require Import\n Events.",
"From ExtLib Require Import\n Core."
] | hoare_example/Imp.v | Var_coerce | |
Lit_coerce : nat -> expr := Lit. | Definition | hoare_example | [
"From Coq Require Import\n Arith.",
"From ExtLib Require Import\n Data.",
"From ITree Require Import\n Events.",
"From ExtLib Require Import\n Core."
] | hoare_example/Imp.v | Lit_coerce | |
Var_coerce : string >-> expr. | Coercion | hoare_example | [
"From Coq Require Import\n Arith.",
"From ExtLib Require Import\n Data.",
"From ITree Require Import\n Events.",
"From ExtLib Require Import\n Core."
] | hoare_example/Imp.v | Var_coerce | |
Lit_coerce : nat >-> expr. Bind Scope expr_scope with expr. Infix "+" := Plus : expr_scope. Infix "-" := Minus : expr_scope. Infix "*" := Mult : expr_scope. Bind Scope stmt_scope with stmt. | Coercion | hoare_example | [
"From Coq Require Import\n Arith.",
"From ExtLib Require Import\n Data.",
"From ITree Require Import\n Events.",
"From ExtLib Require Import\n Core."
] | hoare_example/Imp.v | Lit_coerce | |
ImpState : Type -> Type := | GetVar (x : var) : ImpState value | SetVar (x : var) (v : value) : ImpState unit. | Variant | hoare_example | [
"From Coq Require Import\n Arith.",
"From ExtLib Require Import\n Data.",
"From ITree Require Import\n Events.",
"From ExtLib Require Import\n Core."
] | hoare_example/Imp.v | ImpState | |
denote_aexp (e : aexp) : itree eff value := match e with | AId v => trigger (GetVar v) | ANum n => ret n | APlus a b => l <- denote_aexp a ;; r <- denote_aexp b ;; ret (l + r) | AMinus a b => l <- denote_aexp a ;; r <- denote_aexp b ;; ret (l - r) | AMult a b => l <- denote_aexp a ;; r <- denote_aexp b ;; ret (l * r) e... | Fixpoint | hoare_example | [
"From Coq Require Import\n Arith.",
"From ExtLib Require Import\n Data.",
"From ITree Require Import\n Events.",
"From ExtLib Require Import\n Core."
] | hoare_example/Imp.v | denote_aexp | |
denote_bexp (e : bexp) : itree eff bool := match e with | BTrue => ret true | BFalse => ret false | BEq a b => l <- denote_aexp a ;; r <- denote_aexp b ;; ret (Nat.eqb l r) | BLe a b => l <- denote_aexp a ;; r <- denote_aexp b ;; ret (Nat.leb l r) | BNot a => b <- denote_bexp a ;; ret (negb b) | BAnd a b => l <- denote... | Fixpoint | hoare_example | [
"From Coq Require Import\n Arith.",
"From ExtLib Require Import\n Data.",
"From ITree Require Import\n Events.",
"From ExtLib Require Import\n Core."
] | hoare_example/Imp.v | denote_bexp | |
while (step : itree eff (unit + unit)) : itree eff unit := @iter _ _ _ Iter_Kleisli _ _ (fun _ => step) tt. (** The meaning of statements is now easy to define. They are all straightforward, except for [While], which uses our new [while] combinator over the computation that evaluates the conditional, and then the body ... | Definition | hoare_example | [
"From Coq Require Import\n Arith.",
"From ExtLib Require Import\n Data.",
"From ITree Require Import\n Events.",
"From ExtLib Require Import\n Core."
] | hoare_example/Imp.v | while | |
denote_com (s : com) : itree eff unit := match s with | CAss x e => v <- denote_aexp e ;; trigger (SetVar x v) | CSeq a b => denote_com a ;; denote_com b | CIf i t e => b <- denote_bexp i ;; if (b: bool) then denote_com t else denote_com e | CWhile t body => while (b <- denote_bexp t ;; if (b: bool) then denote_com bod... | Fixpoint | hoare_example | [
"From Coq Require Import\n Arith.",
"From ExtLib Require Import\n Data.",
"From ITree Require Import\n Events.",
"From ExtLib Require Import\n Core."
] | hoare_example/Imp.v | denote_com | |
fact (n:nat): com := input ::= n;;; output ::= 1;;; WHILE ~(n = 0) DO output ::= output * input;;; input ::= input - 1 END. (** We have given _a_ notion of denotation to [fact 6] via [denote_com]. However this is naturally not actually runnable yet, since it contains uninterpreted [ImpState] events. We therefore now ne... | Definition | hoare_example | [
"From Coq Require Import\n Arith.",
"From ExtLib Require Import\n Data.",
"From ITree Require Import\n Events.",
"From ExtLib Require Import\n Core."
] | hoare_example/Imp.v | fact | |
handle_ImpState {E: Type -> Type} `{mapE var 0 -< E}: ImpState ~> itree E := fun _ e => match e with | GetVar x => lookup_def x | SetVar x v => insert x v end. (** We now concretely implement this environment using ExtLib's finite maps. *) | Definition | hoare_example | [
"From Coq Require Import\n Arith.",
"From ExtLib Require Import\n Data.",
"From ITree Require Import\n Events.",
"From ExtLib Require Import\n Core."
] | hoare_example/Imp.v | handle_ImpState | |
env := alist var value. (** Finally, we can define an evaluator for our statements. To do so, we first denote them, leading to an [itree ImpState unit]. We then [interp]ret [ImpState] into [mapE] using [handle_ImpState], leading to an [itree (mapE var value) unit]. Finally, [interp_map] interprets the latter [itree] in... | Definition | hoare_example | [
"From Coq Require Import\n Arith.",
"From ExtLib Require Import\n Data.",
"From ITree Require Import\n Events.",
"From ExtLib Require Import\n Core."
] | hoare_example/Imp.v | env | |
interp_imp {E A} (t : itree (ImpState +' E) A) : stateT env (itree E) A := let t' := interp (bimap handle_ImpState (id_ E)) t in interp_map t'. | Definition | hoare_example | [
"From Coq Require Import\n Arith.",
"From ExtLib Require Import\n Data.",
"From ITree Require Import\n Events.",
"From ExtLib Require Import\n Core."
] | hoare_example/Imp.v | interp_imp | |
eval_imp (s: com) : itree void1 (env * unit) := interp_imp (denote_com s) empty. (** Equipped with this evaluator, we can now compute. Naturally since Coq is total, we cannot do it directly inside of it. We can either rely on extraction, or use some fuel. *) (* YZ Is [burn] broken? *) (* Compute (burn 200 (eval_imp (fa... | Definition | hoare_example | [
"From Coq Require Import\n Arith.",
"From ExtLib Require Import\n Data.",
"From ITree Require Import\n Events.",
"From ExtLib Require Import\n Core."
] | hoare_example/Imp.v | eval_imp | |
E := (ImpState +' E'). (** This interpreter is compatible with the equivalence-up-to-tau. *) Global Instance eutt_interp_imp {R}: Proper (@eutt E R R eq ==> eq ==> @eutt E' (prod (env) R) (prod _ R) eq) interp_imp. Proof. repeat intro. unfold interp_imp. unfold interp_map. rewrite H0. eapply eutt_interp_state_eq; auto.... | Notation | hoare_example | [
"From Coq Require Import\n Arith.",
"From ExtLib Require Import\n Data.",
"From ITree Require Import\n Events.",
"From ExtLib Require Import\n Core."
] | hoare_example/Imp.v | E | |
interp_imp_bind : forall {R S} (t: itree E R) (k: R -> itree E S) (g : env), (interp_imp (ITree.bind t k) g) ≅ (ITree.bind (interp_imp t g) (fun '(g', x) => interp_imp (k x) g')). Proof. intros. unfold interp_imp. unfold interp_map. repeat rewrite interp_bind. repeat rewrite interp_state_bind. apply eqit_bind. reflexiv... | Lemma | hoare_example | [
"From Coq Require Import\n Arith.",
"From ExtLib Require Import\n Data.",
"From ITree Require Import\n Events.",
"From ExtLib Require Import\n Core."
] | hoare_example/Imp.v | interp_imp_bind | |
denote_imp (c : com) : stateT env Delay unit := interp_imp (denote_com c). | Definition | hoare_example | [
"From ExtLib Require Import\n Data.",
"From ITree.Extra Require Import\n Dijkstra.",
"From Paco Require Import paco.",
"From hoare Require Import Imp."
] | hoare_example/ImpHoare.v | denote_imp | |
hoare_triple (P Q : env -> Prop) (c : com) : Prop := forall (s s' :env), P s -> (denote_imp c s ≈ ret (s',tt)) -> Q s'. | Definition | hoare_example | [
"From ExtLib Require Import\n Data.",
"From ITree.Extra Require Import\n Dijkstra.",
"From Paco Require Import paco.",
"From hoare Require Import Imp."
] | hoare_example/ImpHoare.v | hoare_triple | |
lift_imp_post (P : env -> Prop) : Delay (env * unit) -> Prop := fun (t : Delay (env * unit) ) => (exists (s : env), ret (s, tt) ≈ t /\ P s). | Definition | hoare_example | [
"From ExtLib Require Import\n Data.",
"From ITree.Extra Require Import\n Dijkstra.",
"From Paco Require Import paco.",
"From hoare Require Import Imp."
] | hoare_example/ImpHoare.v | lift_imp_post | |
is_bool (E : Type -> Type) (bc : bool) (be : bexp) (s : env) : Prop := @interp_imp E bool (denote_bexp be) s ≈ ret (s, bc). | Definition | hoare_example | [
"From ExtLib Require Import\n Data.",
"From ITree.Extra Require Import\n Dijkstra.",
"From Paco Require Import paco.",
"From hoare Require Import Imp."
] | hoare_example/ImpHoare.v | is_bool | |
is_true (b : bexp) (s : env) : Prop := is_bool void1 true b s. | Definition | hoare_example | [
"From ExtLib Require Import\n Data.",
"From ITree.Extra Require Import\n Dijkstra.",
"From Paco Require Import paco.",
"From hoare Require Import Imp."
] | hoare_example/ImpHoare.v | is_true | |
is_false (b : bexp) (s : env) : Prop := is_bool void1 false b s. (* | Definition | hoare_example | [
"From ExtLib Require Import\n Data.",
"From ITree.Extra Require Import\n Dijkstra.",
"From Paco Require Import paco.",
"From hoare Require Import Imp."
] | hoare_example/ImpHoare.v | is_false | |
unf_intep := unfold interp_imp, interp_map, interp_state, interp, Basics.iter, MonadIter_stateT0, interp, Basics.iter, MonadIter_stateT0. *) | Ltac | hoare_example | [
"From ExtLib Require Import\n Data.",
"From ITree.Extra Require Import\n Dijkstra.",
"From Paco Require Import paco.",
"From hoare Require Import Imp."
] | hoare_example/ImpHoare.v | unf_intep | |
aexp_term : forall (E : Type -> Type) (ae : aexp) (s : env), exists (n : nat), @interp_imp void1 _ (denote_aexp ae) s ≈ Ret (s,n). Proof. intros. induction ae. - exists n. cbn. tau_steps. reflexivity. (*getvar case, extract to a lemma*) - cbn. exists (lookup_default x 0 s). tau_steps. reflexivity. - basic_solve. exists... | Lemma | hoare_example | [
"From ExtLib Require Import\n Data.",
"From ITree.Extra Require Import\n Dijkstra.",
"From Paco Require Import paco.",
"From hoare Require Import Imp."
] | hoare_example/ImpHoare.v | aexp_term | |
bools_term : forall (be : bexp) (s : env), exists (bc : bool), @interp_imp void1 _ (denote_bexp be) s ≈ Ret (s,bc). Proof. intros. induction be. - exists true. cbn. unfold interp_imp, interp_map, interp_state. repeat rewrite interp_ret. tau_steps. reflexivity. - exists false. tau_steps. reflexivity. - specialize (aexp_... | Lemma | hoare_example | [
"From ExtLib Require Import\n Data.",
"From ITree.Extra Require Import\n Dijkstra.",
"From Paco Require Import paco.",
"From hoare Require Import Imp."
] | hoare_example/ImpHoare.v | bools_term | |
classic_bool : forall (b : bexp) (s : env), is_true b s \/ is_false b s. Proof. intros. specialize (bools_term b s) as Hbs. basic_solve. destruct bc; auto. Qed. (* *) | Lemma | hoare_example | [
"From ExtLib Require Import\n Data.",
"From ITree.Extra Require Import\n Dijkstra.",
"From Paco Require Import paco.",
"From hoare Require Import Imp."
] | hoare_example/ImpHoare.v | classic_bool | |
hoare_seq : forall (c1 c2 : com) (P Q R : env -> Prop), {{P}} c1 {{Q}} -> {{Q}} c2 {{R}} -> {{P}} c1 ;;; c2 {{R}}. Proof. unfold hoare_triple. intros c1 c2 P Q R Hc1 Hc2 s s' Hs Hs'. unfold denote_imp in Hs'. cbn in Hs'. rewrite interp_imp_bind in Hs'. fold (denote_imp c1) in Hs'. fold (denote_imp c2) in Hs'. destruct ... | Lemma | hoare_example | [
"From ExtLib Require Import\n Data.",
"From ITree.Extra Require Import\n Dijkstra.",
"From Paco Require Import paco.",
"From hoare Require Import Imp."
] | hoare_example/ImpHoare.v | hoare_seq | |
hoare_if : forall (c1 c2 : com) (b : bexp) (P Q : env -> Prop), {{fun s => is_true b s /\ P s}} c1 {{Q}} -> {{fun s => is_false b s /\ P s}} c2 {{Q}} -> {{ P }} TEST b THEN c1 ELSE c2 FI {{Q}}. Proof. unfold hoare_triple. intros c1 c2 b P Q Hc1 Hc2 s s' Hs. unfold denote_imp. cbn. destruct (classic_bool b s). - unfold ... | Lemma | hoare_example | [
"From ExtLib Require Import\n Data.",
"From ITree.Extra Require Import\n Dijkstra.",
"From Paco Require Import paco.",
"From hoare Require Import Imp."
] | hoare_example/ImpHoare.v | hoare_if |
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